Higher order derivatives
Summary
- If \(f\) is a function with derivative \(f'\) then the derivative of \(f'\) , denoted by \(f''\) , is called the second derivative of \(f\) .
- Example: If \(f\left( x \right)=x^3\) then \(f'\left( x \right)=3x^2\) and \(f^{''}\left( x \right)=6x\) .
- Alternative notation. If \(y=f\left( x \right)=x^3\) then the following are equivalent notation:
- \(f^{''}\left( x \right)=6x\)
- \( \frac{d^2f\left( x \right)}{dx^2}=6x\) or \(d^2f(x)/dx^2=6x\)
- \( \frac{d^2f}{dx^2}\left( x \right)=6x\)
- \( \frac{d^2}{dx^2}f\left( x \right)=6x\) . We can write \( \frac{d^2}{dx^2}x^2=6x\)
- \( \frac{d^2y}{dx^2}=6x\) or \(d^2y/dx^2=6x\)
- \(y^{''}=6x\)
- Sometimes you also see \( \frac{d^2f}{dx^2}=6x\) and \(y^{''}\left( x \right)=6x\) but avoid this notation.
- Higher order derivatives are defined similarly. The n ’ th derivative of \(f\) , \(f\) differentiated \(n\) times is denoted
- \(f^{\left( n \right)}(x) \frac{d^nf\left( x \right)}{dx^n} \frac{d^n}{dx^n}f(x) \frac{d^ny}{dx^n} y^{(n)}\)
- For example, if \(f\left( x \right)=x^3\) then \(f^{\left( 3 \right)}\left( x \right)=6\) and \(f^{\left( n \right)}\left( x \right)=0\) for \(n>3\) .
- For convenience, the zero’th derivative \(f^{(0)}(x)\) is defined as \(f(x)\) .