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Calculus 2 : First and Second Derivatives of Functions

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Example Questions

Example Question #301 : Derivative Review

Find the derivative of the function:

$f(x)=\sqrt[4]{xe^x}$

Possible Answers:

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+{\sqrt[4]{xe^x}}$

$\frac{e^x+1}{4\sqrt[4]{(xe^x)^3}}$

$\frac{e^x}{4\sqrt{(xe^x)^3}}+\frac{{\sqrt[4]{xe^x}}}{4}$

Correct answer:

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

Note that the radical acts as the "outer" function using the first rule, the chain rule.

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Example Question #301 : Derivatives

Find the derivative of the function:

f(x)=3^x+x^2

Possible Answers:

$3^x\ln3+2x$

$3^x\ln3+x^2$

$x^3\ln3+2x$

$\ln3+2x$

Correct answer:

$3^x\ln3+2x$

Explanation:

The derivative of the function is equal to

$f'(x)=3^x\ln3+2x$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} n^x=n^x \ln(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #303 : Derivative Review

Find the derivative of the function:

$f(x)=\ln(\sqrt{x+3}), x\geq-2$

Possible Answers:

$\frac{x+3}{2}$

$\frac{1}{2}$

$\frac{1}{2x+3}$

$\frac{1}{2x+6}$

Correct answer:

$\frac{1}{2x+6}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{1}{2x+6}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #304 : Derivative Review

Find the derivative of the function:

$f(x)=(\cos(x)+e^x)^4$

Possible Answers:

$4(\cos(x)+e^x)^3(\sin(x)+e^x)$

$4(\cos(x)+e^x)^3-\sin(x)+e^x$

$4(\cos(x)+e^x)^3$

$4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

Correct answer:

$4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

Explanation:

The derivative of the function is equal to

$f'(x)=4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x)=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

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Example Question #305 : Derivative Review

Find the derivative of the function:

$f(x)=\csc(4x^2)$

Possible Answers:

$-\left |\csc(4x^2)+\cot(4x^2) \right |$

$-8x\csc(4x^2)\cot(4x^2)$

$8x\csc(4x^2)\cot(4x^2)$

$-\csc(4x^2)\cot(4x^2)$

Correct answer:

$-8x\csc(4x^2)\cot(4x^2)$

Explanation:

The derivative of the function is equal to

$f'(x)=-8x\csc(4x^2)\cot(4x^2)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \csc(x)=-\csc(x)\cot(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #302 : Derivative Review

Find the first derivative of the function:

$f(x)=x\ln(x)+e^{2x}$

Possible Answers:

$\ln(x)+1+e^{2x}$

$\ln(x)+2e^{2x}$

$\ln(x)+x+2e^{2x}$

$\ln(x)+1+2e^{2x}$

Correct answer:

$\ln(x)+1+2e^{2x}$

Explanation:

The derivative of the function is equal to

$f'(x)=\ln(x)+1+2e^{2x}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$

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Example Question #103 : First And Second Derivatives Of Functions

Find the second derivative of f(x)=sinx+tanx.

Possible Answers:

f''(x)=secxtanx+sinx

f''(x)=2sec^2xtanx-sinx

f''(x)=cos(x)

f''(x)=2sec^2xtanx+sinx

f''(x)=secxtanx-sinx

Correct answer:

f''(x)=2sec^2xtanx-sinx

Explanation:

To find the second derivative of a function, we must first find the first derivative.

f(x)=sinx+tanx.

f'(x)=cosx+sec^2x. Now, we must differentiate this function. The first part is a straightforward trigonometric derivative. For the second term, we must use the chain rule.

$f''(x)=-sinx+2secx*\frac{d(secx)}{dx}$

=-sinx+2secx*secxtanx=-sinx+2sec^2xtanx.

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Example Question #104 : First And Second Derivatives Of Functions

Find the derivative of $e^{3x^2}$ .

Possible Answers:

$6x*e^{3x^2}$

$ln(6x)*e^{3x^2}$

$3x^2*e^{3x^2}$

$ln(3x^2)*e^{3x^2}$

$e^{6x}$

Correct answer:

$6x*e^{3x^2}$

Explanation:

The derivative of e^x is special because it returns back the original function (e^x) .

Therefore, this derivative is simply a chain rule application of that exponential function.

Recall the chain rule is,

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

Applying this rule to the original function results in the following derivative.

$\frac{d}{dx}e^{3x^2}=e^u\frac{du}{dx}=e^{3x^2}*6x.$

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Example Question #105 : First And Second Derivatives Of Functions

Find the second derivative of f(x)=sinx-cos(x).

Possible Answers:

sin^2(x)

tan(x)

sinx+cosx

sinx-cosx

cosx-sinx

Correct answer:

cosx-sinx

Explanation:

To get to the second derivative, we must first take the first derivative!

f(x)=sinx-cosx.

f'(x)=cosx--sinx=cosx+sinx.

Then, we will differentiate that to get our answer.

f''(x)=-sinx+cosx.

Recall the trigonometric derivatives:

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

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Example Question #302 : Derivatives

What is the derivative of f(x)=9x^2-5x+1 ?

Possible Answers:

f{}'(x)=18x

$f{}'(x)=18x-5x^2$

f{}'(x)=x-5

f{}'(x)=18

f{}'(x)=18x-5

Correct answer:

f{}'(x)=18x-5

Explanation:

When taking the derivative, multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent. Therefore, the answer is: f{}'(x)=18x-5.

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Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #301 : Derivative Review

Example Question #301 : Derivatives

Example Question #303 : Derivative Review

Example Question #304 : Derivative Review

Example Question #305 : Derivative Review

Example Question #302 : Derivative Review

Example Question #103 : First And Second Derivatives Of Functions

Example Question #104 : First And Second Derivatives Of Functions

Example Question #105 : First And Second Derivatives Of Functions

Example Question #302 : Derivatives

All Calculus 2 Resources

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