Definition
\(\lim\limits_{x\to x_0^+}\samefrac{f(x)-f(x_0)}{x-x_0}=f'(x_0^+)\)
right hand derivative on x₀ point of f function
\(\lim\limits_{x\to x_0^-}\samefrac{f(x)-f(x_0)}{x-x_0}=f'(x_0^-)\)
left hand derivative on x₀ point of f function
If \(f'(x_0^+)=f'(x_0^-)\) derivative of f function can be taken on x₀ point.
\(\lim\limits_{x\to x_0}\samefrac{f(x)-f(x_0)}{x-x_0}=f'(x_0)\)
derivative on x₀ point of f function
Note
\(f'(x)=\samefrac{df(x)}{dx}=\samefrac{dy}{dx}\)
Property 1
\(f(x)=x^n\Rightarrow f'(x)=n\cdot x^{n-1}\)
1.
\(f(x)=x^3\)
\(\Rightarrow f'(x)=?\)
2.
\(f(x)=x^7\)
\(\Rightarrow f'(x)=?\)
3.
\(f(x)=\samefrac1x\)
\(\Rightarrow f'(x)=?\)
4.
\(f(x)=-\samefrac1{x^2}\)
\(\Rightarrow f'(x)=?\)
Property 2
\(f(x)=\sqrt[n]{x^k}=x^{\left(\samefrac kn\right)}\)
\(f'(x)=\samefrac kn\cdot x^{\left(\samefrac kn-1\right)}\)
1.
\(f(x)=\sqrt x\)
\(\Rightarrow f'(x)=?\)
2.
\(f(x)=\sqrt[3] x\)
\(\Rightarrow f'(x)=?\)
3.
\(f(x)=\sqrt[4] x\)
\(\Rightarrow f'(x)=?\)
4.
\(f(x)=-\sqrt x\)
\(\Rightarrow f'(4)=?\)
5.
\(f(x)=\sqrt[5] x\)
\(\Rightarrow f'(-1)=?\)
6.
\(f(x)=\sqrt[3] x-\sqrt x\)
\(\Rightarrow f'(1)=?\)
7.
\(f(x)=\sqrt[7]{x^4}\)
\(\Rightarrow f'(-1)=?\)