Derivatives/Integrals/Trig Identities

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Last updated 8:45 PM on 9/1/26
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33 Terms

1
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Exponential Rule Derivative

\frac{d}{\differentialD x}b^{x}=b^{x}\ln\left(b\right)

2
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Exponential Rule Derivative Chain Rule Application

\frac{d}{\differentialD x}=b^{x}\ln\left(b\right)\cdot f^{\prime}\left(x\right)

3
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Logarithmic Rule Derivative

ddxlog⁡ax=1xln⁡(a)\frac{d}{dx}\log_{a}x=\frac{1}{x\ln\left(a\right)}

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Logarithmic Rule Derivative Chain Rule Application

\frac{d}{\differentialD x}\log_{a}\left\lbrack f\left(x\right)\right\rbrack=\frac{1}{f\left(x\right)\cdot\ln\left(a\right)}\cdot f^{\prime}\left(x\right)=\frac{f^{\prime}\left(x\right)}{f\left(x\right)\cdot\ln\left(a\right)}

5
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\frac{d}{\differentialD x}\sin x=

cos⁡x\cos x

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\frac{d}{\differentialD x}\cos x=

−sin⁡x-\sin x

7
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\frac{d}{\differentialD x}\tan x=

sec⁡2x\sec^2x

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\frac{d}{\differentialD x}\cot x=

−csc⁡2x-\csc^2x^{}

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\frac{d}{\differentialD x}\sec x=

sec⁡xtan⁡x\sec x\tan x

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\frac{d}{\differentialD x}\csc x=

−csc⁡x(cot⁡x)-\csc x\left(\cot x\right)

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\frac{d}{\differentialD x}\ln x=

1x\frac{1}{x}

12
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\frac{d}{\differentialD x}\left(e^{x}\right)=

exe^{x}

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\frac{d}{\differentialD x}\left(\sin^{-1}x\right)=

11−x2\frac{1}{\sqrt{1-x^2}}

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\frac{d}{\differentialD x}\left(\tan^{-1}x\right)=

11+x2\frac{1}{1+x^2}

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\frac{d}{\differentialD x}\left(\sec^{-1}x\right)=

1∣x∣x2−1\frac{1}{\left\vert x\right\vert\sqrt{x^2-1}}

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\frac{d}{\differentialD x}\left(\operatorname{cosec}^{-1}x\right)=

−1∣x∣x2−1-\frac{1}{\left\vert x\right\vert\sqrt{x^2-1}}

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\frac{d}{\differentialD x}\left(\cot^{-1}x\right)=

−11+x2-\frac{1}{1+x^2}

18
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Quotient Rule Derivative

\frac{d}{\differentialD x}f\left\lbrack g\left(x\right)\right\rbrack=\frac{f^{\prime}\left(x\right)\cdot g\left(x\right)-g^{\prime}\left(x\right)\cdot f\left(x\right)}{\left\lbrack g\left(x\right)\right\rbrack^2}

19
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Product Rule Derivative

\frac{d}{\differentialD x}\left\lbrack f\left(x\right)\cdot g\left(x\right)\right\rbrack=f^{\prime}\left(x\right)\cdot g\left(x\right)+g^{\prime}\left(x\right)\cdot f\left(x\right)

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Chain Rule Derivative

\frac{d}{\differentialD x}f\left\lbrack g\left(x\right)\right\rbrack=f^{\prime}\left\lbrack g\left(x\right)\right\rbrack\cdot g^{\prime}\left(x\right)

21
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Pythagorean Identity 1:

sin⁡2x+cos⁡x=1\sin^2x+\cos x=1

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Pythagorean Identity 2:

cos⁡2x=1−sin⁡2x\cos^2x=1-\sin^2x

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Pythagorean Identity 3:

sin⁡2x=1−cos⁡2x\sin^2x=1-\cos^2x

24
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Power Reduction Formula for sin²

1−cos⁡(2θ)2\frac{1-\cos\left(2\theta\right)}{2}

25
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Power Reduction Formula for cos²

1+cos⁡(2θ)2\frac{1+\cos\left(2\theta\right)}{2}

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Power Reduction Formula for tan²

1−cos⁡(2θ)1+cos⁡(2θ)\frac{1-\cos\left(2\theta\right)}{1+\cos\left(2\theta\right)}

27
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∫ ⁣cos⁡x dx\int_{}^{}\!\cos x\,dx =

sin⁡x+C\sin x+C

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∫ ⁣sin⁡x dx=\int_{}^{}\!\sin x\,dx=

−cos⁡x+C-\cos x+C

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∫ ⁣sec⁡2x dx=\int_{}^{}\!\sec^2x\,dx=

tan⁡x+C\tan x+C

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∫ ⁣ex dx=\int_{}^{}\!e^{x}\,dx=

ex+Ce^{x}+C

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∫ ⁣ax dx=\int_{}^{}\!a^{x}\,dx=

axln⁡(a)+C\frac{a^{x}}{\ln\left(a\right)}+C

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∫ ⁣1x dx=\int_{}^{}\!\frac{1}{x}\,dx=

ln⁡∣x∣+C\ln\left\vert x\right\vert+C

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∫ ⁣11+x2 dx=\int_{}^{}\!\frac{1}{1+x^2}\,dx=

tan⁡−1(x)+C\tan^{-1}\left(x\right)+C