Rules for differentiation

Summary

Derivatives of special functions

• Derivative of a constant. If $f(x)=c$ then $f$ is differentiable on $R$ with derivative $f'(x)=0$ for all constants $c$ ,

$$f\left( x \right)=c⟹f'(x)=0$$

• Derivative of $x$ . If $f(x)=x$ then $f$ is differentiable on $R$ with derivative $f'(x)=1$ ,

$$f(x)=x⟹f'(x)=1$$

• Derivative of a linear function. The linear function $f(x)=ax+b$ is differentiable on $R$ with derivative $f'(x)=a$ for all constants $a,b$ ,

$$f(x)=ax+b⟹f'(x)=a$$

• Natural power rule. If $f(x)=x^n$ where $n$ is a natural number then $f$ is differentiable on $R$ with derivative $f'(x)=nx^{n-1}$ ,

$$f(x)=x^n⟹f'(x)=nx^{n-1}$$

• Derivative of a quadratic function. The quadratic function $f(x)=ax^2+bx+c$ is differentiable on $R$ with derivative $f'(x)=2ax+b$ for all constants $a≠0,b,c$ ,

$$f(x)=ax^2+bx+c⟹f'(x)=2ax+b$$

• Derivative of $\frac{1}{x}$ . If $f(x)= \frac{1}{x}$ then $f$ is differentiable on $R\0$ with derivative $f'(x)=- \frac{1}{x^2}$ ,

$$f(x)= \frac{1}{x}⟹f'(x)=- \frac{1}{x^2}$$

• Integer power rule. If $f(x)= \frac{1}{x^n}$ where $n$ is a natural number then $f$ is differentiable on $R\0$ with derivative $f'(x)=- \frac{n}{x^{n+1}}$ ,

$$f(x)= \frac{1}{x^n}⟹f'(x)=- \frac{n}{x^{n+1}}$$

• Derivative of $\sqrt{x}$ . If $f(x)=\sqrt{x}$ then $f$ is differentiable on $(0,∞)$ ( $f$ is not differentiable at $x=0$ ) with derivative $f'(x)= \frac{1}{2\sqrt{x}}$ ,

$$f(x)=\sqrt{x}⟹f'(x)= \frac{1}{2\sqrt{x}}$$

• Real power rule. If $f(x)=x^r$ where $r$ is a real number, $r≠1$ then $f'(x)=rx^{r-1}$ . The domain of $f'$ is $[0,∞)$ if $r≥1$ and $\left( 0,∞ \right)$ if $r<1$ .

$$f(x)=x^r⟹f'(x)=rx^{r-1}$$

• Derivative of $e^x$ . If $f(x)=e^x$ then $f$ is differentiable on $R$ with derivative $f'(x)=e^x$ ,

$$f(x)=e^x⟹f'(x)=e^x$$

• Derivative of $log x$ . If $f(x)=log x$ then $f$ is differentiable on $(0,∞)$ with derivative $f'(x)= \frac{1}{x}$ ,

$$f(x)=log x⟹f'(x)= \frac{1}{x}$$

Derivatives of general functions: DGF

• Additive constant. $c+f$ is differentiable at $x$ ,

$$\frac{d}{dx}\left( c+f\left( x \right) \right)=f'(x)$$

• Multiplicative constant. $cf$ is differentiable at $x$ ,

$$\frac{d}{dx}\left( cf\left( x \right) \right)=cf'(x)$$

• Addition law. $f+g$ is differentiable at $x$ ,

$$\frac{d}{dx}\left( f\left( x \right)+g\left( x \right) \right)=f'(x)+g'(x)$$

• Subtraction law. $f-g$ is differentiable at $x$ ,

$$\frac{d}{dx}\left( f\left( x \right)-g\left( x \right) \right)=f'(x)-g'(x)$$

• Multiplication law. $f⋅g$ is differentiable at $x$ ,

$$\frac{d}{dx}\left( f\left( x \right)g\left( x \right) \right)=f'(x)g(x)+f(x)g'(x)$$

• Division law. If $g(x)≠0$ then $f/g$ is differentiable at $x$ ,

$$\frac{d}{dx} \frac{f(x)}{g(x)}= \frac{f'(x)⋅g(x)-f(x)⋅g'(x)}{g(x)^2}$$