Wize University Calculus 1 Textbook > Derivatives

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Inverse Hyperbolic Functions and their DerivativesPrevious Section

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

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\displaystyle\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}f′(x)=h→0lim​hf(x+h)−f(x)​

=lim⁡h→0[(x+h)2−1]−[x2−1]h=\displaystyle\lim_{h\rightarrow0}\frac{[(x+h)^2-1]-[x^2-1]}{h}=h→0lim​h[(x+h)2−1]−[x2−1]​
                          =

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Find the derivative of each function

a)  y=cos⁡(3)y=\cos\left(3\right)y=cos(3)       b)   y=x3+π4y=\frac{x^3+\pi}{4}y=4x3+π​     c)  y=xe+exy=x^e+e^xy=xe+ex

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Practice Question: Finding a Derivative
Find the derivative of f(x)=x3×xf\left(x\right)=x^3\times\sqrt{x}f(x)=x3×x​.

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 Using the definition of a derivative, find the derivative of  

f(x)=xf(x)=\sqrt{x}f(x)=x​

Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})y=sin(x​)

sin⁡(x)2x\frac{\sin(\sqrt{x})}{2\sqrt{x}}2x​sin(x​)​

cos⁡(x)2x\frac{\cos(\sqrt{x})}{2\sqrt{x}}2x​cos(x​)​

sin⁡(x)2\frac{\sin(\sqrt{x})}{2}2sin(x​)​

cos⁡(x)2\frac{\cos(\sqrt{x})}{2}2cos(x​)​

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 Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}y=esin(x​)

esin⁡(x)cos⁡(x)e^{\sin(\sqrt{x})}{\cos(\sqrt{x})}esin(x​)cos(x​)

esin⁡(x)cos⁡(x)xe^{\sin(\sqrt{x})}\frac{\cos(\sqrt{x})}{\sqrt{x}}esin(x​)x​cos(x​)​

esin⁡(x)cos⁡(x)2xe^{\sin(\sqrt{x})}\frac{\cos(\sqrt{x})}{2\sqrt{x}}esin(x​)2x​cos(x​)​

esin⁡(x)cos⁡(x)xe^{\sin(\sqrt{x})}{\cos(\sqrt{x})}{\sqrt{x}}esin(x​)cos(x​)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 y=2x5−4x(4−x2)3\displaystyle y=\frac{2x^5-4x}{\left(4-x^2\right)^3}y=(4−x2)32x5−4x​, at x=0x=0x=0.

Enter the exact value of the derivative at the point

x=0

I don't know

Find the derivative of  f(x)=x2+3x4/3+1x2+1\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}f(x)=x2+1x2+3x4/3+1​

4x13−2x73(x2+1)2\displaystyle \frac{4x^{\frac{1}{3}}-2x^{\frac{7}{3}}}{\left(x^2+1\right)^2}(x2+1)24x31​−2x37​​

1+2x131+\frac{2}{x^{\frac{1}{3}}}1+x31​2​

4x3+4x+10x73+4x13(x2+1)2\frac{4x^3+4x+10x^{\frac{7}{3}}+4x^{\frac{1}{3}}}{\left(x^2+1\right)^2}(x2+1)24x3+4x+10x37​+4x31​​

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Practice: Power of a Function Rule
Find the derivative of f(x)=(1x+2x3)4f\left(x\right)=\left(\frac{1}{x}+2x^3\right)^4f(x)=(x1​+2x3)4 at the point x=1x=1x=1

Enter the exact value.

f'(1)=

I don't know

Find the derivative of the following:  
yex+y2=x2ye^{x+y^2}=x^2yex+y2=x2

y′=2x+yex+y2ex+y2−2y2ex+y2y' = \frac{2x+ye^{x+y^2}}{e^{x+y^2}-2y^2e^{x+y^2}}y′=ex+y2−2y2ex+y22x+yex+y2​

y′=2x−yex+y2ex+y2−2y2ex+y2y' = \frac{2x-ye^{x+y^2}}{e^{x+y^2}-2y^2e^{x+y^2}}y′=ex+y2−2y2ex+y22x−yex+y2​

y′=2x+yex+y2ex+y2+2y2ex+y2y' = \frac{2x+ye^{x+y^2}}{e^{x+y^2}+2y^2e^{x+y^2}}y′=ex+y2+2y2ex+y22x+yex+y2​

y′=2x−yex+y2ex+y2+2y2ex+y2y' = \frac{2x-ye^{x+y^2}}{e^{x+y^2}+2y^2e^{x+y^2}}y′=ex+y2+2y2ex+y22x−yex+y2​

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Given x2+y=y2−4x^2+y=y^2-4x2+y=y2−4, find y′y'y′.

y' =

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Practice: Second Derivative
Find the second derivative of y=2+x3y=\sqrt{2+x^3}y=2+x3​ at x=1x=1x=1

Enter the exact value of the second derivative at

x=1

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

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If f(x)=xxf(x)=x^{\sqrt{x}}f(x)=xx​ then find f′(4)f'(4)f′(4).

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

−1(1−(sin⁡−1x)2)(1−x2)\frac{-1}{\sqrt{\left(1-(\sin^{-1}x)^2\right)(1-x^2)}}(1−(sin−1x)2)(1−x2)​−1​

−1−x2(1−(sin⁡−1x)2)\frac{-1-x^2}{\sqrt{\left(1-(\sin^{-1}x)^2\right)}}(1−(sin−1x)2)​−1−x2​

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

−1(1−(x)2)(1−x2)\frac{-1}{\sqrt{\left(1-(x)^2\right)(1-x^2)}}(1−(x)2)(1−x2)​−1​

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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 ?

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

Answer

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Find all value(s) of x on the graph of f(x)=1x+16xf\left(x\right)=\frac{1}{x}+16xf(x)=x1​+16x  where the tangent line is horizontal

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Suppose g(x)g\left(x\right)g(x) is differentiable, f(x)=x2g(x)f\left(x\right)=x^2g\left(x\right)f(x)=x2g(x). Given that f′(3)=30f'\left(3\right)=30f′(3)=30, f′′(3)=19f''\left(3\right)=19f′′(3)=19, g(3)=2g\left(3\right)=2g(3)=2, what is g′(3)g'\left(3\right)g′(3) and g′′(3)g''\left(3\right)g′′(3)?

g'(3)=

g''(3)=

I don't know

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)?

Differentiate f(x)=(x3/2)(2x4−5x2−x−1)(xe−π3.5)\displaystyle f(x)=(x^{3/2})\left(2x^{4}-5x^{2}-x^{-1}\right)(x^{e}-\pi^{3.5})f(x)=(x3/2)(2x4−5x2−x−1)(xe−π3.5)

2x112+e−5x72+e−x12+e2x^{\frac{11}{2}+e}-5x^{\frac{7}{2}+e}-x^{\frac{1}{2}+e}2x211​+e−5x27​+e−x21​+e

32x12(8x3−10x+1x2)(exe−1)\frac{3}{2}x^{\frac{1}{2}}\left(8x^3-10x+\frac{1}{x^2}\right)\left(ex^{e-1}\right)23​x21​(8x3−10x+x21​)(exe−1)

(11x92−352x52−12x−12)(xe−π3.5)+(2x112−572−x12)(exe−1)\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)(11x29​−235​x25​−21​x−21​)(xe−π3.5)+(2x211​−527​−x21​)(exe−1)

I don't know

Practice: Normal Line
Find the equation of the normal line to the curve y=2x+xy=2x+\sqrt{x}y=2x+x​ at the point where the normal line has a slope of −13-\frac{1}{3}−31​.

Enter the equation of the normal line in the form

y=mx+b

I don't know

Wize University Calculus 1 Textbook > Derivatives

Exam-Like Practice Questions

Popular Courses

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 $g\left(x\right)$ 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 $g''\left(3\right)$ ?

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 $g'\left(1\right)$ ?

Practice: Normal Line

Wize University Calculus 1 Textbook > Derivatives

Exam-Like Practice Questions

Popular Courses

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 $g'\left(1\right)$ ?