# Differentiation

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### Differentiation of the composite function

$$\frac{d}{dx}x^{n}=nx^{n-1}$$

$$\frac{d}{dx}x=1$$

$$\frac{d}{dx}C=0$$ 'C' is constant.

$$\frac{d}{dx}log_ex=\frac{1}{x}$$

$$\frac{d}{dx}e^{x}=e^{x}$$

### Differentiation formula of the trigonometric function

$$\frac{d}{dx}cos \hspace{1mm}x=-sin \hspace{1mm}x$$

$$\frac{d}{dx}sin \hspace{1mm}x=cos \hspace{1mm}x$$

$$\frac{d}{dx}tan \hspace{1mm}x=sec^{2} \hspace{1mm}x$$

$$\frac{d}{dx}cot \hspace{1mm}x=-cosec^{2} \hspace{1mm}x$$

$$\frac{d}{dx}sec \hspace{1mm}x=sec \hspace{1mm}x\hspace{1mm}tan\hspace{1mm}x$$

$$\frac{d}{dx}cosec \hspace{1mm}x=-cosec \hspace{1mm}x\hspace{1mm}cot\hspace{1mm}x$$

### Differentiation of Hyperbolic functions

$$\frac {d}{dx}sinh\space x = cosh \space x$$

$$\frac {d}{dx}cosh\space x = sinh \space x$$

$$\frac {d}{dx}tanh\space x = sech^2 \space x$$

$$\frac {d}{dx}coth\space x = -cosech^2 \space x$$

$$\frac {d}{dx}sech\space x = -sech \space x \space tanh \space x$$

$$\frac {d}{dx}cosech\space x = -cosech \space x \space coth \space x$$

### Differentiation of Inverse Trignometric functions

$$\frac{d}{dx}sin^{-1}x=\frac{1}{\sqrt{1-x^{2}}}$$

$$\frac{d}{dx}cos^{-1}x=-\frac{1}{\sqrt{1-x^{2}}}$$

$$\frac{d}{dx}tan^{-1}x=\frac{1}{1+x^{2}}$$

$$\frac{d}{dx}cot^{-1}x=-\frac{1}{1+x^{2}}$$

$$\frac{d}{dx}sec^{-1}x=\frac{1}{x\sqrt{x^{2}-1}}$$

$$\frac{d}{dx}cosec^{-1}x=-\frac{1}{x\sqrt{x^{2}-1}}$$

### Logarithmic Differentiation

$$\frac{d}{dx}\frac{a^{x}}{log_ea}=\frac{a^xlog_ea}{log_ea}=a^x$$

$$\frac{d}{dx}a^{x}=a^xlog_ea$$

$$\frac{d}{dx}log_ex=\frac{1}{x}$$

$$\frac{d}{dx}log_ax=\frac{1}{x}log_ae$$

### The formula of Two Functions Differentiation

$$\frac{d}{dx}(f_1(x)\hspace{1mm}f_2(x))= f_1(x)\frac{d}{dx}f_2(x)+f_2(x)\frac{d}{dx}f_1(x)$$

$$\frac{d}{dx}\left(\frac{f_1(x)}{f_2(x)}\right )= \frac{f_2(x)\frac{d}{dx}f_1(x)-f_1(x)\frac{d}{dx}f_2(x)}{\left ( f_2(x) \right )^2 }$$

### Three functions Differentiation

$$\frac{d}{dx}(f_1(x)\space f_2(x)\space f_3(x))= f_2(x)f_3(x)\frac{d}{dx}f_1(x)+f_1(x)f_3(x)\frac{d}{dx}f_2(x)+f_1(x)f_2(x)\frac{d}{dx}f_3(x)$$