Calculus I - Derivatives of common functions

21 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. ddxex\frac{d}{dx}e^xdxd​ex

   ddxex=ex\frac{d}{dx}e^x=e^xdxd​ex=ex

2. ddxsin⁡(x)\frac{d}{dx}\sin(x)dxd​sin(x)

   ddxsin⁡(x)=cos⁡(x)\frac{d}{dx}\sin(x)=\cos(x)dxd​sin(x)=cos(x)

3. ddxcos⁡(x)\frac{d}{dx}\cos(x)dxd​cos(x)

   ddxcos⁡(x)=−sin⁡(x)\frac{d}{dx}\cos(x)=-\sin(x)dxd​cos(x)=−sin(x)

4. ddxtan⁡(x)\frac{d}{dx}\tan(x)dxd​tan(x)

   ddxtan⁡(x)=sec⁡2(x)\frac{d}{dx}\tan(x)=\sec^2(x)dxd​tan(x)=sec2(x)

5. ddxcot⁡(x)\frac{d}{dx}\cot(x)dxd​cot(x)

   ddxcot⁡(x)=−csc⁡2(x)\frac{d}{dx}\cot(x)=-\csc^2(x)dxd​cot(x)=−csc2(x)

6. ddxsec⁡(x)\frac{d}{dx}\sec(x)dxd​sec(x)

   ddxsec⁡(x)=sec⁡(x)tan⁡(x)\frac{d}{dx}\sec(x)=\sec(x)\tan(x)dxd​sec(x)=sec(x)tan(x)

7. ddxcsc⁡(x)\frac{d}{dx}\csc(x)dxd​csc(x)

   ddxcsc⁡(x)=−csc⁡(x)cot⁡(x)\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)dxd​csc(x)=−csc(x)cot(x)

8. ddxln⁡(x)\frac{d}{dx}\ln(x)dxd​ln(x)

   ddxln⁡(x)=1x\frac{d}{dx}\ln(x)=\frac{1}{x}dxd​ln(x)=x1​ , for 0"> x>0x>0x>0 ddxln⁡∣x∣=1x\frac{d}{dx}\ln|x|=\frac{1}{x}dxd​ln∣x∣=x1​ , for x≠0x \neq 0x=0

9. ddxln⁡∣u(x)∣\frac{d}{dx}\ln|u(x)|dxd​ln∣u(x)∣

   ddxln⁡∣u(x)∣=u′(x)u(x)\frac{d}{dx}\ln|u(x)|=\frac{u'(x)}{u(x)}dxd​ln∣u(x)∣=u(x)u′(x)​

10. ddxbx\frac{d}{dx}b^xdxd​bx , for 0"> x>0x>0x>0

    ddxbx=bxln⁡(b)\frac{d}{dx}b^x=b^{x}\ln(b)dxd​bx=bxln(b)

11. ddxg(x)h(x)\frac{d}{dx}g(x)^{h(x)}dxd​g(x)h(x)

    ddxg(x)h(x)=ddxeh(x)ln⁡(g(x))\frac{d}{dx}g(x)^{h(x)}=\frac{d}{dx}e^{h(x)\ln(g(x))}dxd​g(x)h(x)=dxd​eh(x)ln(g(x))

12. ddxlog⁡bx\frac{d}{dx}\log_{b}{x}dxd​logb​x

    ddxlog⁡bx=1xln⁡(b)\frac{d}{dx}\log_{b}{x}=\frac{1}{x\ln(b)}dxd​logb​x=xln(b)1​ , for 0"> x>0x>0x>0 ddxlog⁡b∣x∣=1xln⁡(b)\frac{d}{dx}\log_{b}{|x|}=\frac{1}{x\ln(b)}dxd​logb​∣x∣=xln(b)1​ , for x≠0x \neq 0x=0

13. ddxsin⁡−1(x)\frac{d}{dx}\sin^{-1}(x)dxd​sin−1(x)

    ddxsin⁡−1(x)=11−x2\frac{d}{dx}\sin^{-1}(x)=\frac{1}{\sqrt{1-x^2}}dxd​sin−1(x)=1−x2 ​1​ ,for −1<x<1-1<x<1−1<x<1

14. ddxcos⁡−1(x)\frac{d}{dx}\cos^{-1}(x)dxd​cos−1(x)

    ddxcos⁡−1(x)=−11−x2\frac{d}{dx}\cos^{-1}(x)=-\frac{1}{\sqrt{1-x^2}}dxd​cos−1(x)=−1−x2 ​1​ ,for −1<x<1-1<x<1−1<x<1

15. ddxtan⁡−1(x)\frac{d}{dx}\tan^{-1}(x)dxd​tan−1(x)

    ddxtan⁡−1(x)=11+x2\frac{d}{dx}\tan^{-1}(x)=\frac{1}{1+x^2}dxd​tan−1(x)=1+x21​ , for x∈Rx \in \mathbb{R}x∈R

16. ddxcot⁡−1(x)\frac{d}{dx}\cot^{-1}(x)dxd​cot−1(x)

    ddxcot⁡−1(x)=−11+x2\frac{d}{dx}\cot^{-1}(x)=-\frac{1}{1+x^2}dxd​cot−1(x)=−1+x21​ , for x∈Rx \in \mathbb{R}x∈R

17. ddxsec⁡−1(x)\frac{d}{dx}\sec^{-1}(x)dxd​sec−1(x)

    ddxsec⁡−1(x)=1∣x∣x2−1\frac{d}{dx}\sec^{-1}(x)=\frac{1}{|x|\sqrt{x^2-1}}dxd​sec−1(x)=∣x∣x2−1 ​1​ , for 1"> ∣x∣>1|x|>1∣x∣>1

18. ddxcsc⁡−1(x)\frac{d}{dx}\csc^{-1}(x)dxd​csc−1(x)

    ddxcsc⁡−1(x)=−1∣x∣x2−1\frac{d}{dx}\csc^{-1}(x)=-\frac{1}{|x|\sqrt{x^2-1}}dxd​csc−1(x)=−∣x∣x2−1 ​1​ , for 1"> ∣x∣>1|x|>1∣x∣>1

19. ddxf−1(x)\frac{d}{dx}f^{-1}(x)dxd​f−1(x)

    ddxf−1(x)=1f′(x)\frac{d}{dx}f^{-1}(x)=\frac{1}{f'(x)}dxd​f−1(x)=f′(x)1​ , for f′(x)≠0f'(x) \neq 0f′(x)=0

20. h(x)=f−1(x)h(x)=f^{-1}(x)h(x)=f−1(x) ddxh(x)\frac{d}{dx}h(x)dxd​h(x)

    ddxh(x)=1f′(h(x))\frac{d}{dx}h(x)=\frac{1}{f'(h(x))}dxd​h(x)=f′(h(x))1​ , for f′(h(x))≠0f'(h(x)) \neq 0f′(h(x))=0

21. ddx1g(x)\frac{d}{dx} \frac{1}{g(x)}dxd​g(x)1​

    ddx1g(x)=−g′(x)g(x)2\frac{d}{dx} \frac{1}{g(x)}=\frac{-g'(x)}{g(x)^2}dxd​g(x)1​=g(x)2−g′(x)​