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Calculus 2 : First and Second Derivatives of Functions

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Example Question #281 : Derivatives

Find the derivative of f(x)=4x^2-5x+3 .

Possible Answers:

$f{}'(x)=8x^2$

f'(x)=0

f{}'(x)=8x

$f{}'(x)=8x-\frac{5}{x}$

f{}'(x)=8x-5

Correct answer:

f{}'(x)=8x-5

Explanation:

Recall that when taking the derivative, multiply the exponent of the x term by the coefficient in front and then subtract one from the exponent.

Therefore, your answer is

f{}'(x)=8x-5 .

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Example Question #282 : Derivatives

Find the derivative of x^2+y^2=1 .

Possible Answers:

$\frac{dy}{dx}=-\frac{2x}{y}$

$\frac{dy}{dx}=\frac{x}{y}$

$\frac{dy}{dx}=-\frac{x}{2y}$

$\frac{dy}{dx}=-\frac{x}{y}$

$\frac{dy}{dx}=1$

Correct answer:

$\frac{dy}{dx}=-\frac{x}{y}$

Explanation:

To find the derivative here, you have to use implicit differentiation.

The first step is to take the derivative of each term using the power rule,

$\frac{d}{dx}x^n=nx^{n-1}$

in this particular case it would look like this:

$2x+2y\frac{dy}{dx}=0$ .

Then, solve for $\frac{dy}{dx}$ .

Therefore, your answer is:

$\frac{dy}{dx}=-\frac{x}{y}$ .

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Example Question #282 : Derivatives

Find the first derivative of f(x)=4x^2+2x+1

Possible Answers:

None of the Above

-8x+2

8x-2

8x+2

Correct answer:

8x+2

Explanation:

Step 1: Recall the derivative rules:

-For any term with an exponent, the exponent drops and gets multiplied to the coefficient of that term. The new exponent is one less than the original one..
-For any term in the form "ax", the derivative of this term is just the coefficient.
-For any term with no x term (constant), the derivative is always

Step 2: Take the derivative:

$f'(x)=(4\cdot 2)x^{2-1}+2(1)+0=8x+2$

The first derivative of f(x) is 8x+2

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Example Question #283 : Derivatives

Find the first derivative of $f(x)=\frac {1}{x^3}$

Possible Answers:

$-\frac {3}{x^2}$

$\frac {3}{x^3}$

$-\frac {3}{x^4}$

$\frac {3}{x^4}$

Correct answer:

$-\frac {3}{x^4}$

Explanation:

Step 1: Re-write the function in terms of a term, not a fraction:

$\frac {1}{x^3}=x^{-3}$

Step 2: Using the power rule, the drops down and is put in front of the x term. The exponent gets subtracted by , which means -3-1=-4 . is the new exponent..

$-3x^{-4}$

Step 3: Re-write the exponents with positive numbers...

When we change the exponents to positive from negative, you flip the exponent term over . We can rewrite $x^{-4}$ as $\frac {1}{x^4}$ .

Step 4: Multiply the coefficient from Step and the fraction in step to get the derivative:

$-3 \cdot \frac {1}{x^4}=-\frac {3}{x^4}$

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Example Question #282 : Derivatives

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

Possible Answers:

$2x3^{x^2}ln(3)$

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

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

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

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

Correct answer:

$2x3^{x^2}ln(3)$

Explanation:

This is a special derivative.

$\frac{d}{dx}a^u=a^ulna\frac{du}{dx}.$

$\frac{d}{dx}3^{x^2}=3^{x^2}*ln3*\frac{d(x^2)}{dx}=3^{x^2}ln3*2x.$

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Example Question #286 : Derivatives

What is the derivative of the following function?

$f(x)=10^{sinx}$ .

Possible Answers:

$10^{sinx}*ln(sinx)$

$10^{cosx}*ln(10)$

$10^{sinx}*ln(10)*cos(x)$

$10^{sinx}*ln(10)$

$10^{sinx}*ln(10)*x$

Correct answer:

$10^{sinx}*ln(10)*cos(x)$

Explanation:

The derivative of a constant raised to a power is given by the following equation.

$\frac{d}{dx}a^u=a^ulna\frac{du}{dx}$

$\frac{d}{dx}10^{sinx}=10^{sinx}ln(10)\frac{d(sinx)}{dx}=10^{sinx}*ln(10)*cosx.$

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Example Question #285 : Derivatives

Find the derivative of $f(x)=3x^{2/3}+8x^{-1/2}$

Possible Answers:

$f'(x)=2x^{-1/3}+8x^{1/2}$

$f'(x)=2x^{-1/3}-4x^{1/2}$

$f'(x)=2x^{-1/3}+4x^{1/2}$

$f'(x)=2x^{-1/3}-4x^{-3/2}$

$f'(x)=2x^{-1/3}-8x^{-3/2}$

Correct answer:

$f'(x)=2x^{-1/3}-4x^{-3/2}$

Explanation:

This derivative is an application of the power rule. Remember, we need to multiply the coefficient by the exponent, then reduce the exponent by one.

$f(x)=3x^{2/3}+8x^{-1/2}$

$f'(x)=3*\frac{2}{3}x^{2/3-1}+8*\frac{-1}{2}x^{-1/2-1}=2x^{-1/3}-4x^{-3/2}.$

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Example Question #288 : Derivatives

Find the derivative of the function:

$f(x)=\cos(e^{5x})$

Possible Answers:

$-5e^{5x}\sin(e^{5x})$

$-e^{5x}\sin(e^{5x})$

$-5e^{4x}\sin(e^{5x})$

$5e^{5x}\sin(e^{5x})$

Correct answer:

$-5e^{5x}\sin(e^{5x})$

Explanation:

The derivative of the function is equal to

$f'(x)=-5e^{5x}\sin(e^{5x})$

and was found using the following rules:

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

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Example Question #289 : Derivatives

Find the derivative of the function:

$f(x)=\sec(x^2+x+10)$

Possible Answers:

$\sec(x^2+x+10)\tan(x^2+x+10)$

$(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

$(2x+1)\ln\left |\sec(x^2+x+10)+\tan(x^2+x+10) \right |$

$-(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

Correct answer:

$(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

Explanation:

The derivative of the function is equal to

$f'(x)=(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

and was found using the following rules:

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

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Example Question #290 : Derivatives

Find the derivative of the function:

$f(x)=\frac{x}{e^x}$

Possible Answers:

$\frac{1-x}{e^x}$

$\frac{1+x}{e^x}$

$\frac{1}{e^x}$

$\frac{1-x}{e^{x^2}}$

Correct answer:

$\frac{1-x}{e^x}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{1-x}{e^x}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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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 #281 : Derivatives

Example Question #282 : Derivatives

Example Question #282 : Derivatives

Example Question #283 : Derivatives

Example Question #282 : Derivatives

Example Question #286 : Derivatives

Example Question #285 : Derivatives

Example Question #288 : Derivatives

Example Question #289 : Derivatives

Example Question #290 : Derivatives

All Calculus 2 Resources

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