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High School Math : Using the Chain Rule

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Example Questions

Example Question #1 : Using The Chain Rule

If y= ln(x^3) what is the slope of the line at x=12 .

Possible Answers:

$\infty$

0.1

0.25

0.5

Correct answer:

0.25

Explanation:

The slope at any point on a line is also equal to the derivative. So first we want to find the derivative function of this function and then evaluate it at x=12 . So, to find the derivative we will need to use the chain rule. The chain rule says

$\frac{d}{dx}} f(g(x)) = f'(g(x))g'(x)$

so if we let f(x)= ln(x) and g(x)=x^3 then

since $f'(x)=\frac{1}{x}}$ and g'(x)=3x^2

$\frac{d}{dx}} f(g(x)) = f'(g(x))g'(x) = \frac{1}{x^3}*3x^2 = \frac{3}{x}$

Therefore we evaluate at x=12 and we get $\frac{3}{12}$ or .

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Example Question #62 : Calculus I — Derivatives

What is the first derivative of (x+6)^2 ?

Possible Answers:

4x+24

x+6

2x+12

2x+13

Correct answer:

2x+12

Explanation:

To solve for the first derivative, we're going to use the chain rule. The chain rule says that when taking the derivative of a nested function, your answer is the derivative of the outside times the derivative of the inside.

Mathematically, it would look like this: $\frac{\mathrm{d} }{\mathrm{d} x}f(g(x))=f'(g(x))*g'(x)$

Plug in our equations.

$\frac{\mathrm{d} }{\mathrm{d} x}(x+6)^2=(2(x+6)^{1})*(1+0)$

$\frac{\mathrm{d} }{\mathrm{d} x}(x+6)^2=(2(x+6))$

$\frac{\mathrm{d} }{\mathrm{d} x}(x+6)^2=2x+12$

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Example Question #2 : Using The Chain Rule

$\frac{\mathrm{d} }{\mathrm{d} x} \sin(2x)=?$

Possible Answers:

$2\cos(2x)$

$-\frac{1}{2}\cos(2x)$

$2\sin(2x)$

$\cos(2x)$

$2\cos(x)$

Correct answer:

$2\cos(2x)$

Explanation:

For this problem we need to use the chain rule: f(g(x))=f'(g(x)*g'(x)

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=\frac{\mathrm{d} }{\mathrm{d} x} \sin(2x)*\frac{\mathrm{d} }{\mathrm{d} x}2x$

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=\cos(2x)*2$

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=2\cos(2x)$

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Example Question #3 : Using The Chain Rule

Find the derivative of the following function:

f(x) = (2x+5)^2

Possible Answers:

f'(x) = 20x

f'(x) = 2x+5

f'(x) = 10x+25

f'(x) = 8x+20

f'(x) = 10x+50

Correct answer:

f'(x) = 8x+20

Explanation:

Use -substitution so that u(x) = 2x+5 .

Then the function f(x) becomes f(x) = u^2 .

By the chain rule, $f'(x) = \frac{du}{dx} \times \frac{df}{du}$ .

We calculate each term using the power rule:

$\frac{du}{dx} = 2$

$\frac{df}{du} = 2u$

Plug in u=2x+5 :

f'(x) = 4u = 4 (2x+5) = 8x+20

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High School Math : Using the Chain Rule

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Using The Chain Rule

Example Question #62 : Calculus I — Derivatives

Example Question #2 : Using The Chain Rule

Example Question #3 : Using The Chain Rule

All High School Math Resources

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