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AP Calculus AB : Derivative rules for sums, products, and quotients of functions

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Example Questions

Example Question #461 : Derivatives

Find the derivative of y=(2x^4+5x)(cos(x))

Possible Answers:

y'=8x^3cos(x)+5cos(x)

y'=-2x^4sin(x)+8x^3cos(x)-5xsin(x)+5cos(x)

y'=2x^4cos(x)-2x^4sin(x)+5xcos(x)-5xsin(x)

None of the other answers

Correct answer:

y'=-2x^4sin(x)+8x^3cos(x)-5xsin(x)+5cos(x)

Explanation:

$\frac{d}{dx}[2x^4+5x]=8x^3+5$

$\frac{d}{dx}[cos(x)]=-sin(x)$

Product rule states: $\frac{d}{du}[f(u) \cdot g(u)]=f'(u)g(u)+g'(u)f(u)$

Therefore:

y'=(8x^3+5)(cos(x))-sin(x)(2x^4+5x)

=-2x^4sin(x)+8x^3cos(x)-5xsin(x)+5cos(x)

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Example Question #294 : Computation Of The Derivative

Use the chain rule to differentiate the following function: $h(x)=sin(x^{2}e^{x})$

Possible Answers:

$h^{'}(x)=-sin(x^{2}e^{x})xe^{x}(2+x)$

$h^{'}(x)=-cos(x^{2}e^{x})xe^{x}(2+x)$

$h^{'}(x)=cos(x^{2}e^{x})xe^{x}(2+x)$

$h^{'}(x)=sin(x^{2}e^{x})xe^{x}(2+x)$

Correct answer:

$h^{'}(x)=cos(x^{2}e^{x})xe^{x}(2+x)$

Explanation:

$h(x)=sin(x^{2}e^{x})$

By the chain rule: $h^{'}(x)=cos(x^{2}e^{x})\frac{\mathrm{d} }{\mathrm{d} x}(x^{2}e^{x})$

Differentiate $(x^{2}e^{x})$ using the product rule: $\frac{\mathrm{d} }{\mathrm{d} x}(x^{2}e^{x})=\frac{\mathrm{d} }{\mathrm{d} x}(x^{2})e^{x}+x^{2}\frac{\mathrm{d} }{\mathrm{d} x}e^{x}=2xe^{x}+x^{2}e^{x}$

Substitute this derivative for $\frac{\mathrm{d} }{\mathrm{d} x}(x^{2}e^{x})$ in the first equation: $h^{'}(x)=cos(x^{2}e^{x})(2xe^{x}+x^{2}e^{x})$

Factor the equation: $h^{'}(x)=cos(x^{2}e^{x})xe^{x}(2+x)$

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Example Question #295 : Computation Of The Derivative

f(p)=cos(2p-1)-(2p^3+4p)

Find f'(p) .

Possible Answers:

f'(p)=sin(2p-1)-sin(2p^3+4p)

f'(p)=cos(2p-1)+sin(2p-1)-6p^2+4

f'(p)=-2(sin(2p-1)+3p^2+2)

f'(p)=-2cos(2p-1)(p^3+3p^2+2p+2)

f'(p) is undefined.

Correct answer:

f'(p)=-2(sin(2p-1)+3p^2+2)

Explanation:

$\frac{d}{dx}[cos(2p-1)]=2 \cdot -sin(2p-1)=-2sin(2p-1)$

$\frac{d}{dx}[2p^3+4p]=6p^2+4$

Therefore:

f'(p)=-2sin(2p-1)-(6p^2+4)

=-2sin(2p-1)-6p^2-4

=-2(sin(2p-1)+3p^2+2)

f'(p)=-2(3p^2+sin(2p-1)+2)

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AP Calculus AB : Derivative rules for sums, products, and quotients of functions

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #461 : Derivatives

Example Question #294 : Computation Of The Derivative

Example Question #295 : Computation Of The Derivative

All AP Calculus AB Resources

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