AP Precalculus Flashcards: Change In Tandem

What this deck covers

This deck focuses on Change In Tandem, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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.

Flashcard 1: What is the formula for $\frac{d}{dx}(\tan(x))$?

Answer: $\sec^2(x)$. Standard derivative formula for tangent function.

Flashcard 2: What is the derivative of $\frac{u}{v}$ using the Quotient Rule?

Answer: $\frac{u'v - uv'}{v^2}$. Quotient rule: top derivative times bottom minus top times bottom derivative, over bottom squared.

Flashcard 3: What is the derivative of $\ln(x^3)$?

Answer: $\frac{3}{x}$. Using logarithm property: $\ln(x^3) = 3\ln(x)$, so derivative is $\frac{3}{x}$.

Flashcard 4: What is the derivative of $\frac{u}{v}$ using the Quotient Rule?

Answer: $\frac{u'v - uv'}{v^2}$. Quotient rule: top derivative times bottom minus top times bottom derivative, over bottom squared.

Flashcard 5: What is the derivative of $\cos(x)$?

Answer: $-\sin(x)$. Standard derivative formula for cosine function.

Flashcard 6: Find the derivative of $e^{3x}$.

Answer: $3e^{3x}$. Chain rule: derivative of $e^u$ is $e^u \cdot u'$, where $u = 3x$.

Flashcard 7: Identify the derivative of $\ln(x)$.

Answer: $\frac{1}{x}$. Standard derivative formula for natural logarithm.

Flashcard 8: Determine $\frac{d}{dx}(\ln(x^2))$.

Answer: $\frac{2}{x}$. Using chain rule with $\ln(u)$ where $u = x^2$, so $\frac{u'}{u} = \frac{2x}{x^2}$.

Flashcard 9: What is the derivative of $\sin^2(x)$ using the Chain Rule?

Answer: $2\sin(x)\cos(x)$. Chain rule with power rule: $2\sin(x) \cdot \cos(x)$.

Flashcard 10: What is the derivative of $\ln(x^3)$?

Answer: $\frac{3}{x}$. Using logarithm property: $\ln(x^3) = 3\ln(x)$, so derivative is $\frac{3}{x}$.

Flashcard 11: Determine $\frac{d}{dx}(\tan(x))$.

Answer: $\sec^2(x)$. Standard derivative formula for tangent function.

Flashcard 12: Identify the derivative of $\sin(g(x))$ using the Chain Rule.

Answer: $\cos(g(x)) \cdot g'(x)$. Derivative of sine is cosine, multiplied by derivative of inner function.

Flashcard 13: Calculate $\frac{d}{dx}(x^5 + x^3)$.

Answer: $5x^4 + 3x^2$. Apply power rule to each term separately.

Flashcard 14: Find the derivative of $x^{\frac{1}{2}}$.

Answer: $\frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2} \cdot x^{-1/2}$.

Flashcard 15: Determine $\frac{d}{dx}(\cot(x))$.

Answer: $-\csc^2(x)$. Standard derivative formula for cotangent function.

Flashcard 16: Calculate $\frac{d}{dx}(5x^4)$.

Answer: $20x^3$. Constant factor rule: multiply derivative by the constant 5.

Flashcard 17: What is the formula for $\frac{d}{dx}(\tan(x))$?

Answer: $\sec^2(x)$. Standard derivative formula for tangent function.

Flashcard 18: What is the derivative of $\tan^{-1}(x)$?

Answer: $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

Flashcard 19: What is the derivative of a constant $c$?

Answer:

1. Constants have zero rate of change.

Flashcard 20: State the Product Rule for differentiation.

Answer: $(uv)' = u'v + uv'$. Derivative of first times second plus first times derivative of second.

Flashcard 21: Determine $\frac{d}{dx}(\sqrt{x})$.

Answer: $\frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ and apply power rule.

Flashcard 22: What is the derivative of $u \cdot v$ using the Product Rule?

Answer: $u'v + uv'$. Apply product rule to find derivative of product of functions.

Flashcard 23: Identify the derivative of $\cos^{-1}(x)$.

Answer: $-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.

Flashcard 24: Find the derivative of $\ln(e^x)$.

Answer: $1$. Since $\ln(e^x) = x$, the derivative is simply 1.

Flashcard 25: State the Power Rule for differentiation.

Answer: $\frac{d}{dx}(x^n) = nx^{n-1}$. Bring down the exponent and reduce the power by one.

Flashcard 26: Identify the derivative of $x^{-1}$.

Answer: $-x^{-2}$. Power rule with negative exponent: $-1 \cdot x^{-2}$

Flashcard 27: Calculate $\frac{d}{dx}(x^5 + x^3)$.

Answer: $5x^4 + 3x^2$. Apply power rule to each term separately.

Flashcard 28: What is the derivative of $x^3$?

Answer: $3x^2$. Power rule: bring down exponent 3, reduce power to 2.

Flashcard 29: What is the derivative of $u \cdot v$ using the Product Rule?

Answer: $u'v + uv'$. Apply product rule to find derivative of product of functions.

Flashcard 30: Calculate $\frac{d}{dx}(\frac{1}{x+2})$.

Answer: $-\frac{1}{(x+2)^2}$. Chain rule: derivative of $u^{-1}$ is $-u^{-2} \cdot u'$.

Flashcard 31: Find the derivative of $\csc(x)$.

Answer: $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

Flashcard 32: What is the formula for the derivative of $f(g(x))$ using the Chain Rule?

Answer: $f'(g(x)) \cdot g'(x)$. Apply outer function derivative at inner function, then multiply by inner derivative.

Flashcard 33: Find the derivative of $\ln(e^x)$.

Answer: $1$. Since $\ln(e^x) = x$, the derivative is simply 1.

Flashcard 34: Find the derivative of $x^{\frac{1}{2}}$.

Answer: $\frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2} \cdot x^{-1/2}$.

Flashcard 35: State the Power Rule for differentiation.

Answer: $\frac{d}{dx}(x^n) = nx^{n-1}$. Bring down the exponent and reduce the power by one.

Flashcard 36: State the Chain Rule for differentiation.

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies derivatives when functions are composed inside each other.

Flashcard 37: What is the derivative of $\sec(x)$?

Answer: $\sec(x)\tan(x)$. Standard derivative formula for secant function.

Flashcard 38: Calculate $\frac{d}{dx}(\cos(5x))$.

Answer: $-5\sin(5x)$. Derivative of cosine is negative sine, times derivative of inner function.

Flashcard 39: Determine $\frac{d}{dx}(\cot(x))$.

Answer: $-\csc^2(x)$. Standard derivative formula for cotangent function.

Flashcard 40: Determine $\frac{d}{dx}(\sqrt{x})$.

Answer: $\frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ and apply power rule.

Flashcard 41: Find the derivative of $\csc(x)$.

Answer: $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

Flashcard 42: Find the derivative of $\frac{1}{x}$.

Answer: $-\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

Flashcard 43: Determine $\frac{d}{dx}(\sin^{-1}(x))$.

Answer: $\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse sine function.

Flashcard 44: What is the formula for the derivative of $f(g(x))$ using the Chain Rule?

Answer: $f'(g(x)) \cdot g'(x)$. Apply outer function derivative at inner function, then multiply by inner derivative.

Flashcard 45: Determine $\frac{d}{dx}(\sin^{-1}(x))$.

Answer: $\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse sine function.

Flashcard 46: What is the derivative of $\sin^2(x)$ using the Chain Rule?

Answer: $2\sin(x)\cos(x)$. Chain rule with power rule: $2\sin(x) \cdot \cos(x)$.

Flashcard 47: What is the derivative of $\tan^{-1}(x)$?

Answer: $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

Flashcard 48: Identify the derivative of $x^{-1}$.

Answer: $-x^{-2}$. Power rule with negative exponent: $-1 \cdot x^{-2}$.

Flashcard 49: What does $\frac{dy}{dx}$ represent in relation to change in tandem?

Answer: Rate of change of $y$ with respect to $x$. Shows how fast $y$ changes as $x$ changes at any point.

Flashcard 50: What is the derivative of $\sin(x)$?

Answer: $\cos(x)$. Standard derivative formula for sine function.

Flashcard 51: State the Product Rule for differentiation.

Answer: $(uv)' = u'v + uv'$. Derivative of first times second plus first times derivative of second.

Flashcard 52: What is the derivative of $e^x$?

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

Flashcard 53: Calculate $\frac{d}{dx}(5x^4)$.

Answer: $20x^3$. Constant factor rule: multiply derivative by the constant 5.

Flashcard 54: What is the derivative of $\sec(x)$?

Answer: $\sec(x)\tan(x)$. Standard derivative formula for secant function.

Flashcard 55: Identify the derivative of $\cos^{-1}(x)$.

Answer: $-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.