AP CALC BC UNIT 3 DERIVATIVES

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Calculus

Derivatives & Differentiation

AP Calculus BC

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Inverse Function Derivatives

HIGH_SCHOOL

Last updated 8:24 PM on 9/9/26
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17 Terms

1
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ef(x)e^{f\left(x\right)}

ef(x)f′(x)e^{f\left(x\right)}f^{\prime}\left(x\right)

2
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af(x)a^{f\left(x\right)}

af(x)ln⁡(a)f′(x)a^{f\left(x\right)}\ln\left(a\right)f^{\prime}\left(x\right)

3
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ln⁡(f(x))\ln\left(f\left(x\right)\right)

f′(x)f(x)\frac{f^{\prime}\left(x\right)}{f\left(x\right)}

4
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log⁡a(f(x))\log_{a}\left(f\left(x\right)\right)

f′(x)f(x)ln⁡(a)\frac{f^{\prime}\left(x\right)}{f\left(x\right)\ln\left(a\right)}

5
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sin⁡(f(x))\sin\left(f\left(x\right)\right)

cos⁡(f(x))f′(x)\cos\left(f\left(x\right)\right)f^{\prime}\left(x\right)

6
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cos⁡(f(x))\cos\left(f\left(x\right)\right)

−sin⁡(f(x))f′(x)-\sin\left(f\left(x\right)\right)f^{\prime}\left(x\right)

7
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tan⁡(f(x))\tan\left(f\left(x\right)\right)

sec⁡2(f(x))(f′(x))\sec^2\left(f\left(x\right)\right)\left(f^{\prime}\left(x\right)\right)

8
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csc⁡(f(x))\csc\left(f\left(x\right)\right)

−csc⁡(f(x))cot⁡(f(x))f′(x)-\csc\left(f\left(x\right)\right)\cot\left(f\left(x\right)\right)f^{\prime}\left(x\right)

9
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sec⁡(f(x))\sec\left(f\left(x\right)\right)

sec⁡(f(x))tan⁡(f(x))f′(x)\sec\left(f\left(x\right)\right)\tan\left(f\left(x\right)\right)f^{\prime}\left(x\right)

10
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cot⁡(f(x))\cot\left(f\left(x\right)\right)

−csc⁡2(f(x))f′(x)-\csc^2\left(f\left(x\right)\right)f^{\prime}\left(x\right)

11
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sin⁡−1(f(x))\sin^{-1}\left(f\left(x\right)\right)

f′(x)1−[f(x)]2\frac{f^{\prime}\left(x\right)}{\sqrt{1-\left\lbrack f\left(x\right)\right\rbrack^2}}

12
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cos⁡−1(f(x))\cos^{-1}\left(f\left(x\right)\right)

−f′(x)1−[f(x)]2-\frac{f^{\prime}\left(x\right)}{\sqrt{1-\left\lbrack f\left(x\right)\right\rbrack^2}}

13
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tan⁡−1(f(x))\tan^{-1}\left(f\left(x\right)\right)

f′(x)1+[f(x)]2\frac{f^{\prime}\left(x\right)}{1+\left\lbrack f\left(x\right)\right\rbrack^2}

14
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csc⁡−1(f(x))\csc^{-1}\left(f\left(x\right)\right)

−f′(x)∣f(x)∣[f(x)]2−1-\frac{f^{\prime}\left(x\right)}{\left|f\left(x\right)\right|\sqrt{\left\lbrack f\left(x\right)\right\rbrack^2-1}}

15
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sec⁡−1(f(x))\sec^{-1}\left(f\left(x\right)\right)

f′(x)∣f(x)∣[f(x)]2−1\frac{f^{\prime}\left(x\right)}{\left|f\left(x\right)\right|\sqrt{\left\lbrack f\left(x\right)\right\rbrack^2-1}}

16
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cot⁡−1(f(x))\cot^{-1}\left(f\left(x\right)\right)

−f′(x)1+[f(x)]2-\frac{f^{\prime}\left(x\right)}{1+\left\lbrack f\left(x\right)\right\rbrack^2}

17
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(f−1)′(x)\left(f^{-1}\right)^{\prime}\left(x\right)

1f′(f−1(x))\frac{1}{f^{\prime}\left(f^{-1}\left(x\right)\right)}