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GRE Subject Test: Math : Chain Rule

Study concepts, example questions & explanations for GRE Subject Test: Math

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Example Questions

Example Question #1 : Chain Rule

Compute the derivative:

y=cos(sin(2x))

Possible Answers:

-2sin^2(2x)

-2cos^2(2x)

-2cos(2x)sin(sin(2x))

-2cos(2x)sin^2(2x)

-2sec(2x)tan(2x)

Correct answer:

-2cos(2x)sin(sin(2x))

Explanation:

This question requires application of multiple chain rules. There are 2 inner functions in y=cos(sin(2x)) , which are sin(2x) and .

The brackets are to identify the functions within the function where the chain rule must be applied.

Solve the derivative.

$\frac{dy}{dx}= -sin([sin(2x)]) \cdot (cos[2x]))\cdot 2=-2cos(2x)sin(sin(2x))$

The sine of sine of an angle cannot be combined to be sine squared.

Therefore, the answer is:

-2cos(2x)sin(sin(2x))

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Example Question #12 : Derivatives & Integrals

Find the derivative of the following function:

$h(x)=(3x^3-4x)^{33}$

Possible Answers:

$h'(x)=(297x^2+132)(3x^2-4x)^{33}$

$h'(x)=(297x^2-132)(3x^2-4x)^{32}$

$h'(x)=(297x^2-132)(3x^2-4x)^{33}$

$h'(x)=(9x^2-4)(3x^2-4x)^{32}$

$h'(x)=(297x^2)(3x^2-4x)^{32}$

Correct answer:

$h'(x)=(297x^2-132)(3x^2-4x)^{32}$

Explanation:

Recall chain rule for this problem

(f(g(x)))'=f'(g(x))g'(x)

So if we are given the following,

$h(x)=(3x^3-4x)^{33}$

We can think of it like this

f(x)=3x^3-4x

f'(x)=9x^2-4

$g(x)=x^{33}$

$g'(x)=33x^{32}$

$(f(g(x)))'=(9x^2-4)(33)(3x^2-4x)^{32}$

Clean it up a bit to get:

$(f(g(x)))'=(297x^2-132)(3x^2-4x)^{32}$

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Example Question #1 : Chain Rule

What is the derivative of 3x^2*8y^3

Possible Answers:

3x^2*24y^2+8y^3*6x

6x*24y

3x^2*24y^2

8y^3*6x

3x*8y^2

Correct answer:

3x^2*24y^2+8y^3*6x

Explanation:

Chain Rule: Given: f(x)*g(y)

$\frac{\mathrm{dy} }{\mathrm{d} x}=f(x)g'(y)+g(y)f'(x)$

For this problem

f(x)=3x^2

$f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}3x^2=3(2)x^{2-1}=6x$

g(y)=8y^3

$g'(y)=\frac{\mathrm{d} }{\mathrm{d} x}8y^3=8(3)y^{3-1}=24y^2$

Plug the values into the Chain Rule formula and simplify:

$\frac{\mathrm{dy} }{\mathrm{d} x}(3x^2*8y^3)=3x^2*24y^2+8y^3*6x$

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GRE Subject Test: Math : Chain Rule

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Chain Rule

Example Question #12 : Derivatives & Integrals

Example Question #1 : Chain Rule

All GRE Subject Test: Math Resources

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