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 \(x\) ,

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