AP Calculus AB Flashcards: Rate Of Change At A Point

Study Rate Of Change At A Point in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Rate Of Change At A Point

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QUESTION

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How do you symbolically represent the derivative of $f(x)$ at $x=a$ ?

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ANSWER

$f'(a)$ . Standard notation for the derivative at a specific point.

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What this deck covers

This deck focuses on Rate Of Change At A Point, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

All flashcards

Flashcard 1: How do you symbolically represent the derivative of $f(x)$ at $x=a$ ?

Answer: $f'(a)$ . Standard notation for the derivative at a specific point.

Flashcard 2: What is the derivative of a constant function?

Answer: Zero. Constants have no rate of change.

Flashcard 3: What does the instantaneous rate of change of a function represent?

Answer: Slope of tangent line. The tangent touches the curve at exactly one point.

Flashcard 4: Compute the instantaneous rate of change for $f(x)=x^3$ at $x=1$ .

Answer:

Using $f'(x) = 3x^2$ , so $f'(1) = 3$ .

Flashcard 5: What is the derivative of $x^n$ using power rule?

Answer: $nx^{n-1}$ . The power rule for differentiation.

Flashcard 6: Find $f'(a)$ for $f(x) = \frac{1}{x}$ at $x=2$ .

Answer: $-\frac{1}{4}$ . Using $f'(x) = -\frac{1}{x^2}$ , so $f'(2) = -\frac{1}{4}$ .

Flashcard 7: Which mathematical concept represents the instantaneous rate of change?

Answer: Derivative. The derivative measures instantaneous rate of change.

Flashcard 8: What is the relationship between average rate of change and secant lines?

Answer: Average rate = Slope of secant line. Secant lines connect two points, giving average rate.

Flashcard 9: Find the derivative of $f(x)=\text{cos}(x)$ .

Answer: $-\text{sin}(x)$ . The derivative of cosine is negative sine.

Flashcard 10: Evaluate the average rate of change of $f(x)=x^3$ from $x=1$ to $x=2$ .

Answer:

$\frac{8-1}{2-1} = 7$

Flashcard 11: Which function has a constant average rate of change?

Answer: Linear function. Linear functions have the same rate everywhere.

Flashcard 12: Calculate $f'(x)$ for $f(x)=\text{ln}(x)$ .

Answer: $\frac{1}{x}$ . The derivative of the natural logarithm function.

Flashcard 13: If $f(x) = x^2 + 3x$ , what is $f'(x)$ ?

Answer: $2x + 3$ . Apply power rule to each term.

Flashcard 14: Find the instantaneous rate of change for $f(x)=x^2+2x$ at $x=3$ .

Answer:

$f'(x) = 2x + 2$ , so $f'(3) = 8$ .

Flashcard 15: What does the average rate of change of a function represent?

Answer: Slope of secant line. Connects two points on the function graph.

Flashcard 16: How is the instantaneous rate of change at a point $x=a$ defined?

Answer: Derivative $f'(a)$ . The derivative gives the instantaneous rate of change at a specific point.

Flashcard 17: Identify the geometric representation of $f'(a)$ .

Answer: Slope of tangent line at $x=a$ . The derivative equals the slope of the line tangent to the curve.

Flashcard 18: Identify the derivative of $f(x)=\text{e}^x$ at $x=0$ .

Answer:

Since $f'(x) = e^x$ and $e^0 = 1$ .

Flashcard 19: What is the relationship between average rate of change and secant lines?

Answer: Average rate = Slope of secant line. Secant lines connect two points, giving average rate.

Flashcard 20: What is the derivative of $f(x)=\tan(x)$ ?

Answer: $\sec^2(x)$ . The derivative of tangent is secant squared.

Flashcard 21: What is the derivative of the constant function $f(x)=c$ ?

Answer:

Constants have no rate of change.

Flashcard 22: Find $f'(a)$ for $f(x) = \frac{1}{x}$ at $x=2$ .

Answer: $-\frac{1}{4}$ . Using $f'(x) = -\frac{1}{x^2}$ , so $f'(2) = -\frac{1}{4}$ .

Flashcard 23: Identify the derivative of $f(x)=\text{e}^x$ at $x=0$ .

Answer:

Since $f'(x) = e^x$ and $e^0 = 1$ .

Flashcard 24: What is the formula for average rate of change of $f(x)$ from $x=a$ to $x=b$ ?

Answer: $\frac{f(b) - f(a)}{b - a}$ . Standard formula for average rate of change over an interval.

Flashcard 25: In which scenario does average rate of change equal instantaneous rate of change?

Answer: When $f(x)$ is linear. Linear functions have constant slope everywhere.

Flashcard 26: If $f(x) = x^2 + 3x$ , what is $f'(x)$ ?

Answer: $2x + 3$ . Apply power rule to each term.

Flashcard 27: Which mathematical concept represents the instantaneous rate of change?

Answer: Derivative. The derivative measures instantaneous rate of change.

Flashcard 28: Calculate $f'(x)$ for $f(x)=\text{ln}(x)$ .

Answer: $\frac{1}{x}$ . The derivative of the natural logarithm function.

Flashcard 29: What does the average rate of change of a function represent?

Answer: Slope of secant line. Connects two points on the function graph.

Flashcard 30: What is $f'(x)$ for $f(x)=e^x$ ?

Answer: $e^x$ . The exponential function is its own derivative.

Flashcard 31: What is the formula for average rate of change of $f(x)$ from $x=a$ to $x=b$ ?

Answer: $\frac{f(b) - f(a)}{b - a}$ . Standard formula for average rate of change over an interval.

Flashcard 32: How do you symbolically represent the derivative of $f(x)$ at $x=a$ ?

Answer: $f'(a)$ . Standard notation for the derivative at a specific point.

Flashcard 33: What is the instantaneous rate of change of $f(x)=5$ at any point?

Answer: $0$ . Constant functions have zero derivative.

Flashcard 34: Find the instantaneous rate of change of $f(x)=x^2$ at $x=2$ .

Answer:

Using $f'(x) = 2x$ , so $f'(2) = 4$ .

Flashcard 35: Which function has a constant average rate of change?

Answer: Linear function. Linear functions have the same rate everywhere.

Flashcard 36: What is the derivative of $f(x)=\tan(x)$ ?

Answer: $\sec^2(x)$ . The derivative of tangent is secant squared.

Flashcard 37: State the formal definition of a derivative at a point $x=a$ .

Answer: $f'(a) = \text{lim}_{h \to 0} \frac{f(a+h) - f(a)}{h}$ . The limit definition of the derivative using difference quotient.

Flashcard 38: What does the instantaneous rate of change of a function represent?

Answer: Slope of tangent line. The tangent touches the curve at exactly one point.

Flashcard 39: What is the geometric interpretation of the derivative?

Answer: Slope of the tangent line. The derivative represents the slope of the tangent line.

Flashcard 40: Calculate the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$ .

Answer:

$\frac{9-1}{3-1} = \frac{8}{2} = 4$

Flashcard 41: What is the interpretation of $f'(x)$ in terms of velocity?

Answer: Velocity function. When position is $f(x)$ , $f'(x)$ represents velocity.

Flashcard 42: Determine $f'(x)$ if $f(x)=\frac{1}{x}$ .

Answer: $-\frac{1}{x^2}$ . Derivative of $x^{-1}$ using the power rule.

Flashcard 43: What is the geometric interpretation of the derivative?

Answer: Slope of the tangent line. The derivative represents the slope of the tangent line.

Flashcard 44: Compute the instantaneous rate of change for $f(x)=x^3$ at $x=1$ .

Answer:

Using $f'(x) = 3x^2$ , so $f'(1) = 3$ .

Flashcard 45: Determine $f'(3)$ for $f(x)=3x^2 + 2x$ .

Answer:

$f'(x) = 6x + 2$ , so $f'(3) = 18 + 2 = 20$ .

Flashcard 46: Identify the geometric representation of $f'(a)$ .

Answer: Slope of tangent line at $x=a$ . The derivative equals the slope of the line tangent to the curve.

Flashcard 47: Evaluate the average rate of change of $f(x)=x^3$ from $x=1$ to $x=2$ .

Answer:

$\frac{8-1}{2-1} = 7$

Flashcard 48: Identify the formula for the difference quotient.

Answer: $\frac{f(a+h) - f(a)}{h}$ . The ratio used in the limit definition of derivatives.

Flashcard 49: What is $f'(x)$ for $f(x)=e^x$ ?

Answer: $e^x$ . The exponential function is its own derivative.

Flashcard 50: Find the instantaneous rate of change of $f(x)=x^2$ at $x=2$ .

Answer:

Using $f'(x) = 2x$ , so $f'(2) = 4$ .

Flashcard 51: Determine the derivative of $f(x) = \text{ln}(x)$ at $x=1$ .

Answer:

Using $f'(x) = \frac{1}{x}$ , so $f'(1) = 1$ .

Flashcard 52: Determine $f'(3)$ for $f(x)=3x^2 + 2x$ .

Answer:

$f'(x) = 6x + 2$ , so $f'(3) = 18 + 2 = 20$ .

Flashcard 53: Calculate the average rate of change of $f(x)=3x+5$ from $x=2$ to $x=5$ .

Answer:

Linear functions have constant slope of 3.

Flashcard 54: What is the derivative of $x^n$ using power rule?

Answer: $nx^{n-1}$ . The power rule for differentiation.

Flashcard 55: State the formal definition of a derivative at a point $x=a$ .

Answer: $f'(a) = \text{lim}_{h \to 0} \frac{f(a+h) - f(a)}{h}$ . The limit definition of the derivative using difference quotient.

Flashcard 56: Calculate the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$ .

Answer:

$\frac{9-1}{3-1} = \frac{8}{2} = 4$

Flashcard 57: Determine $f'(x)$ if $f(x)=\frac{1}{x}$ .

Answer: $-\frac{1}{x^2}$ . Derivative of $x^{-1}$ using the power rule.

Flashcard 58: Find the instantaneous rate of change for $f(x)=x^2+2x$ at $x=3$ .

Answer:

$f'(x) = 2x + 2$ , so $f'(3) = 8$ .

Flashcard 59: What is the derivative of a constant function?

Answer: Zero. Constants have no rate of change.

Flashcard 60: What is the derivative of the constant function $f(x)=c$ ?

Answer:

Constants have no rate of change.

Flashcard 61: If $f(x)=\text{sin}(x)$ , what is $f'(x)$ ?

Answer: $\text{cos}(x)$ . The derivative of sine is cosine.

Flashcard 62: In which scenario does average rate of change equal instantaneous rate of change?

Answer: When $f(x)$ is linear. Linear functions have constant slope everywhere.

Flashcard 63: What is the instantaneous rate of change of $f(x)=5$ at any point?

Answer:

Constant functions have zero derivative.

Flashcard 64: If $f(x)=\text{sin}(x)$ , what is $f'(x)$ ?

Answer: $\text{cos}(x)$ . The derivative of sine is cosine.

Flashcard 65: Identify the formula for the difference quotient.

Answer: $\frac{f(a+h) - f(a)}{h}$ . The ratio used in the limit definition of derivatives.

Flashcard 66: Determine the derivative of $f(x) = \text{ln}(x)$ at $x=1$ .

Answer:

Using $f'(x) = \frac{1}{x}$ , so $f'(1) = 1$ .

Flashcard 67: How is the instantaneous rate of change at a point $x=a$ defined?

Answer: Derivative $f'(a)$ . The derivative gives the instantaneous rate of change at a specific point.

Flashcard 68: Calculate the average rate of change of $f(x)=3x+5$ from $x=2$ to $x=5$ .

Answer:

Linear functions have constant slope of 3.

Flashcard 69: Find the derivative of $f(x)=\cos(x)$ .

Answer: $-\sin(x)$ . The derivative of cosine is negative sine.

Flashcard 70: What is the interpretation of $f'(x)$ in terms of velocity?

Answer: Velocity function. When position is $f(x)$ , $f'(x)$ represents velocity.

AP Calculus AB Flashcards: Rate Of Change At A Point

Rate Of Change At A Point

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: How do you symbolically represent the derivative of f(x)f(x)f(x) at x=ax=ax=a?

Flashcard 2: What is the derivative of a constant function?

Flashcard 3: What does the instantaneous rate of change of a function represent?

Flashcard 4: Compute the instantaneous rate of change for f(x)=x3f(x)=x^3f(x)=x3 at x=1x=1x=1.

Flashcard 5: What is the derivative of xnx^nxn using power rule?

Flashcard 6: Find f′(a)f'(a)f′(a) for f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=2x=2x=2.

Flashcard 7: Which mathematical concept represents the instantaneous rate of change?

Flashcard 8: What is the relationship between average rate of change and secant lines?

Flashcard 9: Find the derivative of f(x)=cos(x)f(x)=\text{cos}(x)f(x)=cos(x).

Flashcard 10: Evaluate the average rate of change of f(x)=x3f(x)=x^3f(x)=x3 from x=1x=1x=1 to x=2x=2x=2.

Flashcard 11: Which function has a constant average rate of change?

Flashcard 12: Calculate f′(x)f'(x)f′(x) for f(x)=ln(x)f(x)=\text{ln}(x)f(x)=ln(x).

Flashcard 13: If f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x, what is f′(x)f'(x)f′(x)?

Flashcard 14: Find the instantaneous rate of change for f(x)=x2+2xf(x)=x^2+2xf(x)=x2+2x at x=3x=3x=3.

Flashcard 15: What does the average rate of change of a function represent?

Flashcard 16: How is the instantaneous rate of change at a point x=ax=ax=a defined?

Flashcard 17: Identify the geometric representation of f′(a)f'(a)f′(a).

Flashcard 18: Identify the derivative of f(x)=exf(x)=\text{e}^xf(x)=ex at x=0x=0x=0.

Flashcard 19: What is the relationship between average rate of change and secant lines?

Flashcard 20: What is the derivative of f(x)=tan⁡(x)f(x)=\tan(x)f(x)=tan(x)?

Flashcard 21: What is the derivative of the constant function f(x)=cf(x)=cf(x)=c?

Flashcard 22: Find f′(a)f'(a)f′(a) for f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=2x=2x=2.

Flashcard 23: Identify the derivative of f(x)=exf(x)=\text{e}^xf(x)=ex at x=0x=0x=0.

Flashcard 24: What is the formula for average rate of change of f(x)f(x)f(x) from x=ax=ax=a to x=bx=bx=b?

Flashcard 25: In which scenario does average rate of change equal instantaneous rate of change?

Flashcard 26: If f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x, what is f′(x)f'(x)f′(x)?

Flashcard 27: Which mathematical concept represents the instantaneous rate of change?

Flashcard 28: Calculate f′(x)f'(x)f′(x) for f(x)=ln(x)f(x)=\text{ln}(x)f(x)=ln(x).

Flashcard 29: What does the average rate of change of a function represent?

Flashcard 30: What is f′(x)f'(x)f′(x) for f(x)=exf(x)=e^xf(x)=ex?

Flashcard 31: What is the formula for average rate of change of f(x)f(x)f(x) from x=ax=ax=a to x=bx=bx=b?

Flashcard 32: How do you symbolically represent the derivative of f(x)f(x)f(x) at x=ax=ax=a?

Flashcard 33: What is the instantaneous rate of change of f(x)=5f(x)=5f(x)=5 at any point?

Flashcard 34: Find the instantaneous rate of change of f(x)=x2f(x)=x^2f(x)=x2 at x=2x=2x=2.

Flashcard 35: Which function has a constant average rate of change?

Flashcard 36: What is the derivative of f(x)=tan⁡(x)f(x)=\tan(x)f(x)=tan(x)?

Flashcard 37: State the formal definition of a derivative at a point x=ax=ax=a.

Flashcard 38: What does the instantaneous rate of change of a function represent?

Flashcard 39: What is the geometric interpretation of the derivative?

Flashcard 40: Calculate the average rate of change of f(x)=x2f(x)=x^2f(x)=x2 from x=1x=1x=1 to x=3x=3x=3.

Flashcard 41: What is the interpretation of f′(x)f'(x)f′(x) in terms of velocity?

Flashcard 42: Determine f′(x)f'(x)f′(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Flashcard 43: What is the geometric interpretation of the derivative?

Flashcard 44: Compute the instantaneous rate of change for f(x)=x3f(x)=x^3f(x)=x3 at x=1x=1x=1.

Flashcard 45: Determine f′(3)f'(3)f′(3) for f(x)=3x2+2xf(x)=3x^2 + 2xf(x)=3x2+2x.

Flashcard 46: Identify the geometric representation of f′(a)f'(a)f′(a).

Flashcard 47: Evaluate the average rate of change of f(x)=x3f(x)=x^3f(x)=x3 from x=1x=1x=1 to x=2x=2x=2.

Flashcard 48: Identify the formula for the difference quotient.

Flashcard 49: What is f′(x)f'(x)f′(x) for f(x)=exf(x)=e^xf(x)=ex?

Flashcard 50: Find the instantaneous rate of change of f(x)=x2f(x)=x^2f(x)=x2 at x=2x=2x=2.

Flashcard 51: Determine the derivative of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) at x=1x=1x=1.

Flashcard 52: Determine f′(3)f'(3)f′(3) for f(x)=3x2+2xf(x)=3x^2 + 2xf(x)=3x2+2x.

Flashcard 53: Calculate the average rate of change of f(x)=3x+5f(x)=3x+5f(x)=3x+5 from x=2x=2x=2 to x=5x=5x=5.

Flashcard 54: What is the derivative of xnx^nxn using power rule?

Flashcard 55: State the formal definition of a derivative at a point x=ax=ax=a.

Flashcard 56: Calculate the average rate of change of f(x)=x2f(x)=x^2f(x)=x2 from x=1x=1x=1 to x=3x=3x=3.

Flashcard 57: Determine f′(x)f'(x)f′(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Flashcard 58: Find the instantaneous rate of change for f(x)=x2+2xf(x)=x^2+2xf(x)=x2+2x at x=3x=3x=3.

Flashcard 59: What is the derivative of a constant function?

Flashcard 60: What is the derivative of the constant function f(x)=cf(x)=cf(x)=c?

Flashcard 61: If f(x)=sin(x)f(x)=\text{sin}(x)f(x)=sin(x), what is f′(x)f'(x)f′(x)?

Flashcard 62: In which scenario does average rate of change equal instantaneous rate of change?

Flashcard 63: What is the instantaneous rate of change of f(x)=5f(x)=5f(x)=5 at any point?

Flashcard 64: If f(x)=sin(x)f(x)=\text{sin}(x)f(x)=sin(x), what is f′(x)f'(x)f′(x)?

Flashcard 65: Identify the formula for the difference quotient.

Flashcard 66: Determine the derivative of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) at x=1x=1x=1.

Flashcard 67: How is the instantaneous rate of change at a point x=ax=ax=a defined?

Flashcard 68: Calculate the average rate of change of f(x)=3x+5f(x)=3x+5f(x)=3x+5 from x=2x=2x=2 to x=5x=5x=5.

Flashcard 69: Find the derivative of f(x)=cos⁡(x)f(x)=\cos(x)f(x)=cos(x).

Flashcard 70: What is the interpretation of f′(x)f'(x)f′(x) in terms of velocity?

Flashcard 1: How do you symbolically represent the derivative of $f(x)$ at $x=a$ ?

Flashcard 4: Compute the instantaneous rate of change for $f(x)=x^3$ at $x=1$ .

Flashcard 5: What is the derivative of $x^n$ using power rule?

Flashcard 6: Find $f'(a)$ for $f(x) = \frac{1}{x}$ at $x=2$ .

Flashcard 9: Find the derivative of $f(x)=\text{cos}(x)$ .

Flashcard 10: Evaluate the average rate of change of $f(x)=x^3$ from $x=1$ to $x=2$ .

Flashcard 12: Calculate $f'(x)$ for $f(x)=\text{ln}(x)$ .

Flashcard 13: If $f(x) = x^2 + 3x$ , what is $f'(x)$ ?

Flashcard 14: Find the instantaneous rate of change for $f(x)=x^2+2x$ at $x=3$ .

Flashcard 16: How is the instantaneous rate of change at a point $x=a$ defined?

Flashcard 17: Identify the geometric representation of $f'(a)$ .

Flashcard 18: Identify the derivative of $f(x)=\text{e}^x$ at $x=0$ .

Flashcard 20: What is the derivative of $f(x)=\tan(x)$ ?

Flashcard 21: What is the derivative of the constant function $f(x)=c$ ?

Flashcard 22: Find $f'(a)$ for $f(x) = \frac{1}{x}$ at $x=2$ .

Flashcard 23: Identify the derivative of $f(x)=\text{e}^x$ at $x=0$ .

Flashcard 24: What is the formula for average rate of change of $f(x)$ from $x=a$ to $x=b$ ?

Flashcard 26: If $f(x) = x^2 + 3x$ , what is $f'(x)$ ?

Flashcard 28: Calculate $f'(x)$ for $f(x)=\text{ln}(x)$ .

Flashcard 30: What is $f'(x)$ for $f(x)=e^x$ ?

Flashcard 31: What is the formula for average rate of change of $f(x)$ from $x=a$ to $x=b$ ?

Flashcard 32: How do you symbolically represent the derivative of $f(x)$ at $x=a$ ?

Flashcard 33: What is the instantaneous rate of change of $f(x)=5$ at any point?

Flashcard 34: Find the instantaneous rate of change of $f(x)=x^2$ at $x=2$ .

Flashcard 36: What is the derivative of $f(x)=\tan(x)$ ?

Flashcard 37: State the formal definition of a derivative at a point $x=a$ .

Flashcard 40: Calculate the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$ .

Flashcard 41: What is the interpretation of $f'(x)$ in terms of velocity?

Flashcard 42: Determine $f'(x)$ if $f(x)=\frac{1}{x}$ .

Flashcard 44: Compute the instantaneous rate of change for $f(x)=x^3$ at $x=1$ .

Flashcard 45: Determine $f'(3)$ for $f(x)=3x^2 + 2x$ .

Flashcard 46: Identify the geometric representation of $f'(a)$ .

Flashcard 47: Evaluate the average rate of change of $f(x)=x^3$ from $x=1$ to $x=2$ .

Flashcard 49: What is $f'(x)$ for $f(x)=e^x$ ?

Flashcard 50: Find the instantaneous rate of change of $f(x)=x^2$ at $x=2$ .

Flashcard 51: Determine the derivative of $f(x) = \text{ln}(x)$ at $x=1$ .

Flashcard 52: Determine $f'(3)$ for $f(x)=3x^2 + 2x$ .

Flashcard 53: Calculate the average rate of change of $f(x)=3x+5$ from $x=2$ to $x=5$ .

Flashcard 54: What is the derivative of $x^n$ using power rule?

Flashcard 55: State the formal definition of a derivative at a point $x=a$ .

Flashcard 56: Calculate the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$ .

Flashcard 57: Determine $f'(x)$ if $f(x)=\frac{1}{x}$ .

Flashcard 58: Find the instantaneous rate of change for $f(x)=x^2+2x$ at $x=3$ .

Flashcard 60: What is the derivative of the constant function $f(x)=c$ ?

Flashcard 61: If $f(x)=\text{sin}(x)$ , what is $f'(x)$ ?

Flashcard 63: What is the instantaneous rate of change of $f(x)=5$ at any point?

Flashcard 64: If $f(x)=\text{sin}(x)$ , what is $f'(x)$ ?

Flashcard 66: Determine the derivative of $f(x) = \text{ln}(x)$ at $x=1$ .

Flashcard 67: How is the instantaneous rate of change at a point $x=a$ defined?

Flashcard 68: Calculate the average rate of change of $f(x)=3x+5$ from $x=2$ to $x=5$ .

Flashcard 69: Find the derivative of $f(x)=\cos(x)$ .

Flashcard 70: What is the interpretation of $f'(x)$ in terms of velocity?