Wize University Calculus 1 Textbook > Derivatives

Exam-Like Practice Questions

Using the definition of derivative, find f′(x) for f(x) = x2 − 1

f'(x)=\displaystyle\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}

=\displaystyle\lim_{h\rightarrow0}\frac{[(x+h)^2-1]-[x^2-1]}{h}

Your answer

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Find the derivative of each function

a) $y=\cos\left(3\right)$ b) $y=\frac{x^3+\pi}{4}$ c) $y=x^e+e^x$

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Practice Question: Finding a Derivative

Find the derivative of $f\left(x\right)=x^3\times\sqrt{x}$ .

I don't know

Using the definition of a derivative, find the derivative of

f(x)=\sqrt{x}

$\frac{1}{\sqrt{x}}$

$2\sqrt{x}$

$\frac{1}{2\sqrt{x}}$

$\sqrt{x}$

I don't know

Find the derivative of the following function:

y=\sin(\sqrt{x})

$\frac{\sin(\sqrt{x})}{2\sqrt{x}}$

$\frac{\cos(\sqrt{x})}{2\sqrt{x}}$

$\frac{\sin(\sqrt{x})}{2}$

$\frac{\cos(\sqrt{x})}{2}$

I don't know

Find the derivative of the following function:

y=e^{\sin(\sqrt{x})}

$e^{\sin(\sqrt{x})}{\cos(\sqrt{x})}$

$e^{\sin(\sqrt{x})}\frac{\cos(\sqrt{x})}{\sqrt{x}}$

$e^{\sin(\sqrt{x})}\frac{\cos(\sqrt{x})}{2\sqrt{x}}$

$e^{\sin(\sqrt{x})}{\cos(\sqrt{x})}{\sqrt{x}}$

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Find the derivative of the following functions at the given point using quotient rule.

*You may also need to use other derivative rules you've learned so far.

Derivative of $\displaystyle y=\frac{2x^5-4x}{\left(4-x^2\right)^3}$ , at $x=0$ .

Enter the exact value of the derivative at the point $x=0$

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Find the derivative of $\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}$

$\displaystyle \frac{4x^{\frac{1}{3}}-2x^{\frac{7}{3}}}{\left(x^2+1\right)^2}$

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

$\frac{4x^3+4x+10x^{\frac{7}{3}}+4x^{\frac{1}{3}}}{\left(x^2+1\right)^2}$

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Practice: Power of a Function Rule

Find the derivative of $f\left(x\right)=\left(\frac{1}{x}+2x^3\right)^4$ at the point $x=1$

Enter the exact value. $f'(1)=$

I don't know

Find the derivative of the following:

ye^{x+y^2}=x^2

$y' = \frac{2x+ye^{x+y^2}}{e^{x+y^2}-2y^2e^{x+y^2}}$

$y' = \frac{2x-ye^{x+y^2}}{e^{x+y^2}-2y^2e^{x+y^2}}$

$y' = \frac{2x+ye^{x+y^2}}{e^{x+y^2}+2y^2e^{x+y^2}}$

$y' = \frac{2x-ye^{x+y^2}}{e^{x+y^2}+2y^2e^{x+y^2}}$

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Given $x^2+y=y^2-4$ , find $y'$ .

y' =

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Practice: Second Derivative

Find the second derivative of $y=\sqrt{2+x^3}$ at $x=1$

Enter the exact value of the second derivative at $x=1$

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Find the derivative of $y=\frac{\left(x-3\right)^9\ x^x\ \ln\ x}{x-9}.$

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If $f(x)=x^{\sqrt{x}}$ then find $f'(4)$ .

$4\ln 4+4$

$4\ln 4+8$

$4\ln 4+2$

$2\ln 4+2$

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Find the derivative of $g(x)=\cos^{-1}\left(\sin^{-1}x\right)$ .

$\frac{-1}{\sqrt{\left(1-(\sin^{-1}x)^2\right)(1-x^2)}}$

$\frac{-1-x^2}{\sqrt{\left(1-(\sin^{-1}x)^2\right)}}$

$\frac{-sinx}{\sqrt{\left(1-(\sin^{-1}x)^2\right)}}$

$\frac{-1}{\sqrt{\left(1-(x)^2\right)(1-x^2)}}$

I don't know

Is there any point on the graph y = x2+ + 3x + 8 such that the tangent line is parallel to the line with the equation y = 7x + 104 ?

( 2,18 )

( 4,20 )

( 3,15 )

( 6,22 )

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Find the equation of tangent line to $y=\sqrt{2x+3}$ at x=3.

Your answer

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Find all value(s) of x on the graph of $f\left(x\right)=\frac{1}{x}+16x$ where the tangent line is horizontal

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Suppose $g\left(x\right)$ is differentiable, $f\left(x\right)=x^2g\left(x\right)$ . Given that $f'\left(3\right)=30$ , $f''\left(3\right)=19$ , $g\left(3\right)=2$ , what is $g'\left(3\right)$ and $g''\left(3\right)$ ?

$g'(3)=$$g''(3)=$

I don't know

Suppose $g\left(x\right)$ is differentiable, $f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}$ , $f'\left(1\right)=2$ , $g\left(1\right)=1$ , what is $g'\left(1\right)$ ?

I don't know

Differentiate $\displaystyle f(x)=(x^{3/2})\left(2x^{4}-5x^{2}-x^{-1}\right)(x^{e}-\pi^{3.5})$

$2x^{\frac{11}{2}+e}-5x^{\frac{7}{2}+e}-x^{\frac{1}{2}+e}$

$\frac{3}{2}x^{\frac{1}{2}}\left(8x^3-10x+\frac{1}{x^2}\right)\left(ex^{e-1}\right)$

$\displaystyle\left(11x^{\frac{9}{2}}-\frac{35}{2}x^{\frac{5}{2}}-\frac{1}{2}x^{-\frac{1}{2}}\right)\left(x^e-\pi^{3.5}\right)+\left(2x^{\frac{11}{2}}-5^{\frac{7}{2}}-x^{\frac{1}{2}}\right)\left(ex^{e-1}\right)$

I don't know

Practice: Normal Line

Find the equation of the normal line to the curve $y=2x+\sqrt{x}$ at the point where the normal line has a slope of $-\frac{1}{3}$ .

Enter the equation of the normal line in the form $y=mx+b$

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Exam-Like Practice Questions

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Practice Question: Finding a Derivative

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Find the derivative of the following functions at the given point using quotient rule.

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Practice: Power of a Function Rule

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Practice: Second Derivative

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Suppose $g\left(x\right)$ is differentiable, $f\left(x\right)=x^2g\left(x\right)$ . Given that $f'\left(3\right)=30$ , $f''\left(3\right)=19$ , $g\left(3\right)=2$ , what is $g'\left(3\right)$ and $g''\left(3\right)$ ?

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Suppose $g\left(x\right)$ is differentiable, $f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}$ , $f'\left(1\right)=2$ , $g\left(1\right)=1$ , what is $g'\left(1\right)$ ?

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Practice: Normal Line

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Exam-Like Practice Questions

Practice Question: Finding a Derivative

Find the derivative of the following functions at the given point using quotient rule.

Practice: Power of a Function Rule

Practice: Second Derivative

Suppose g(x)g\left(x\right)g(x) is differentiable, f(x)=1+x g(x)x2f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}f(x)=x21+x g(x)​, f′(1)=2f'\left(1\right)=2f′(1)=2, g(1)=1g\left(1\right)=1g(1)=1, what is g′(1)g'\left(1\right)g′(1)?

Practice: Normal Line

Suppose $g\left(x\right)$ is differentiable, $f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}$ , $f'\left(1\right)=2$ , $g\left(1\right)=1$ , what is $g'\left(1\right)$ ?