Derivative formulas and rules reference

Product Rule

$$(fg{)}^{\prime }=f^{\prime }g+g^{\prime }f$$

Quotient Rule

$$(\frac{f}{g}{)}^{\prime }=\frac{f^{\prime }g-g^{\prime }f}{g^2}$$

Chain Rule

$$\frac{d}{dx}(f(g(x)))=f^{\prime }(g(x))g^{\prime }(x)$$

Examples

$$\frac{d}{dx}[(f(x){)}^n]=n[f(x){]}^{n-1}{f}^{\prime }(x)$$

$$\frac{d}{dx}(e^{f(x)})=f^{\prime }(x)e^{f(x)}$$

$$\frac{d}{dx}[\ln (f(x))=\frac{f^{\prime }(x)}{f(x)}$$

$$\frac{d}{dx}[\sin (f(x))]=f^{\prime }(x)\cos [f(x)]$$

$$\frac{d}{dx}[\cos (f(x)]=-f^{\prime }(x)\sin [f(x)]$$

$$\frac{d}{dx}[{\tan }^{-1}(f(x))]=f^{\prime }(x){\sec }^2[f(x)]$$

$$\frac{d}{dx}[\sec (f(x))]=f^{\prime }(x)\sec [f(x)]\tan [f(x)]$$

$$\frac{d}{dx}({\tan }^{-1}[f(x)]=\frac{f^{\prime }}{1+[f(x)]}$$

Polynomials

$$\begin{array}{r}\frac{d}{dx}(c)=0\frac{d}{dx}(x)=1\frac{d}{dx}(cx)=c\\ \\ \frac{d}{dx}(x^n)=n{x}^{n-1}\frac{d}{dx}(c{x}^n)=nc{x}^{n-1}\end{array}$$

Trigonometry

$$\begin{array}{rlr}\frac{d}{dx}(\sin x)=\cos x& & \frac{d}{dx}(\sec x)=\sec x\tan x\\ \frac{d}{dx}(\cos x)=-\sin x& & \frac{d}{dx}(\csc x)=-\csc x\cot x\\ \frac{d}{dx}(\tan x)={\sec }^{2}x& & \frac{d}{dx}(\cot x)=-{\csc }^{2}x\end{array}$$

Inverse Trigonometry

$$\begin{array}{rlr}\frac{d}{dx}({\sin }^{-1}x)=\frac{1}{\sqrt{1-x^2}}& & \frac{d}{dx}({\sec }^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}}\\ \frac{d}{dx}({\cos }^{-1}x)=-\frac{1}{\sqrt{1-x^2}}& & \frac{d}{dx}({\csc }^{-1}x)=-\frac{1}{|x|\sqrt{x^2-1}}\\ \frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2}& & \frac{d}{dx}({\cot }^{-1}x)=-\frac{1}{1+x^2}\end{array}$$

Hyperbolic Trigonometry

$$\begin{array}{rlr}\frac{d}{dx}(\sinh x)=\cosh x& & \frac{d}{dx}(\mathrm{sech}x)=-\mathrm{sech}x\tanh x\\ \frac{d}{dx}(\cosh x)=\sinh x& & \frac{d}{dx}(\mathrm{csch}x)=-\mathrm{csch}x\coth x\\ \frac{d}{dx}(\tanh x)={\mathrm{sech}}^{2}x& & \frac{d}{dx}(\coth x)=-{\mathrm{csch}}^{2}x\end{array}$$

Exponential and Logarithmic

$$\begin{array}{rlr}\frac{d}{dx}(e^x)=e^x& & \frac{d}{dx}(a^x)=a^x\ln (a)\\ \frac{d}{dx}(\ln x)=\frac{1}{x}& & \frac{d}{dx}({\log }_a(x))=\frac{1}{x\ln a}\end{array}$$

for the derivative of $\ln (x),x>0$ for $\ln |x|,x\ne 0$

$$\begin{array}{rlr}\frac{d}{dx}(e^{g(x)})=g^{\prime }(x)e^{g(x)}& & \frac{d}{dx}(\ln g(x))=\frac{g^{\prime }(x)}{g(x)}\end{array}$$

this is just chain rule, but shows up quite frequently

Implicit

$$\frac{d}{dx}[x^n{y}^m=c]$$

$$\begin{array}{rl}& \Rightarrow n{x}^{n-1}{y}^m+m{y}^{\prime }{y}^{m-1}{x}^n=0\\ & \Rightarrow y^{\prime }=\frac{-n{x}^{n-1}{y}^m}{m{y}^{m-1}{x}^n}\end{array}$$

---

$$\frac{d}{dx}[x^n+y^m=c]$$

$$\begin{array}{rl}& \Rightarrow n{x}^{n-1}+m{y}^{m-1}{y}^{\prime }=0\\ & \Rightarrow y^{\prime }=\frac{-n{x}^{n-1}}{m{y}^{m-1}}\end{array}$$

---

$$\frac{d}{dx}[e^{y^m}=\sin (x^n)]$$

$$\begin{array}{rl}& \Rightarrow m{y}^{m-1}{y}^{\prime }{e}^{y^m}=n{x}^{n-1}\cos (x^n)\\ & \Rightarrow y^{\prime }=\frac{n{x}^{n-1}\cos (x^n)}{m{y}^{m-1}{e}^{y^m}}\end{array}$$

---

$$\frac{d}{dx}[\sin (y^m)=e^{x^2}]$$

$$\begin{array}{rl}& \Rightarrow m{y}^{m-1}{y}^{\prime }\cos (y^m)=n{x}^{n-1}{e}^{x^n}\\ & \Rightarrow y^{\prime }=\frac{n{x}^{n-1}{e}^{x^n}}{m{y}^{m-1}\cos (y^m)}\end{array}$$