Calculus/Differentiation/Basics of Differentiation/Exercises
< Calculus | Differentiation
Find the Derivative by Definition
Find the derivative of the following functions using the limit definition of the derivative.
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Prove the Constant Rule
10. Use the definition of the derivative to prove that for any fixed real number ,
![{\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e100ac3ba73695d9c36ad8ada81efba4c2a01129)
![{\displaystyle {\begin{aligned}{\frac {d}{dx}}[c\cdot f(x)]&=\lim _{\Delta x\to 0}{\frac {c\cdot f(x+\Delta x)-c\cdot f(x)}{\Delta x}}\\&=c\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\\&=c{\frac {d}{dx}}[f(x)]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c07443c9244e623f4190ad90f2f8dda344f84a57)
Find the Derivative by Rules
Find the derivative of the following functions:
Power Rule
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![{\displaystyle f(x)=3{\sqrt[{3}]{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b861dfb68b951e23d614557d2cb071669b0851)
![{\displaystyle f'(x)={\frac {1}{\sqrt[{3}]{x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94b10c8c5e4cbed34f88eb9719f071d312ee68a5)
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![{\displaystyle f'(x)={\frac {-2}{x^{3}}}+{\frac {1}{\sqrt[{3}]{x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3c4021f5bbdf8fa04027a4fe4c907a8c10cc18)
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![{\displaystyle f(x)={\frac {3}{x^{4}}}-{\sqrt[{4}]{x}}+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2d5e5287d1e1f9e9a3f66fe3d81364fbed337c)
![{\displaystyle f'(x)={\frac {-12}{x^{5}}}-{\frac {1}{4{\sqrt[{4}]{x^{3}}}}}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c20d3197b7b30c1a5a6492e0b36998f19c1dc0)
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![{\displaystyle f'(x)={\frac {2}{\sqrt[{3}]{x^{2}}}}-{\frac {0.4}{x^{0.6}}}-{\frac {18}{x^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afda870d55f5f874bfde34936a0bafe945ccd502)
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![{\displaystyle f(x)={\frac {1}{\sqrt[{3}]{x}}}+{\sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa56b0f9f1e3f9dbd84b8e241f26eec58e0f966e)
![{\displaystyle f'(x)={\frac {-1}{3{\sqrt[{3}]{x^{4}}}}}+{\frac {1}{2{\sqrt {x}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244d0f0859c7036984f53a7fd8ed474028a5afcc)
Product Rule
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Quotient Rule
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Chain Rule
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Exponentials
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Logarithms
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Trigonometric functions
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More Differentiation
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![{\displaystyle {\frac {d}{dx}}[(x^{3}+5)^{10}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/322d22bf641e5d6a35d63be10b39c7353404dfcd)
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![{\displaystyle {\frac {d}{dx}}[x^{3}+3x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/052c718ffadbb9cb4954b7dd5791f055ff1ce77c)
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![{\displaystyle {\frac {d}{dx}}[(x+4)(x+2)(x-3)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/160ae1c70a96f198eb4c1fd16d0ed2940ce4acfb)
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![{\displaystyle {\frac {d}{dx}}[{\frac {x+1}{3x^{2}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9320f75bc1f03266f545ee487bd178d212956699)
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![{\displaystyle {\frac {d}{dx}}[3x^{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d330c0494fcf0513cd524464634f6360c2ed4c23)
58.
![{\displaystyle {\frac {d}{dx}}[x^{4}\sin(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c54697e3e6e8b7be626f8ad40a67ef061ab14bde)
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![{\displaystyle {\frac {d}{dx}}[2^{x}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70c4dd6b628068248913d496b20327629fc271d2)
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![{\displaystyle {\frac {d}{dx}}[e^{x^{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70feadeaf0a4d91cf66a795d8d2f169e942ec523)
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![{\displaystyle {\frac {d}{dx}}[e^{2^{x}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1c7dffc918e0d7462c5f97f9a145db84a16597)
Implicit Differentiation
Use implicit differentiation to find y'
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Logarithmic Differentiation
Use logarithmic differentiation to find :
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![{\displaystyle y=x({\sqrt[{4}]{1-x^{3}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d04c97ee04319439a96db44e9efa9e702123a0)
![{\displaystyle y'={\sqrt[{4}]{1-x^{3}}}+{\frac {3x^{3}}{4(1-x^{3})^{\frac {3}{4}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28e6b605b503ecc19dc8eb0fa9804e1dbfbaeb5c)
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Equation of Tangent Line
For each function, , (a) determine for what values of the tangent line to is horizontal and (b) find an equation of the tangent line to at the given point.
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75. Find an equation of the tangent line to the graph defined by at the point (1,-1).
76. Find an equation of the tangent line to the graph defined by at the point (1,0).
Higher Order Derivatives
77. What is the second derivative of ?
78. Use induction to prove that the (n+1)th derivative of a n-th order polynomial is 0.
