Solved example of definition of derivative
Find the derivative of $x^2$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x^2$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
Square of the first term: $\left(x\right)^2 = [a^2]$.
Double product of the first by the second: $2\left(x\right)\left(h\right) = [2ab]$.
Square of the second term: $\left(h\right)^2 = [b^2]$.
Expand $\left(x+h\right)^2$
Cancel like terms $x^2$ and $-x^2$
Factor the polynomial $2xh+h^2$ by it's GCF: $h$
Simplify the fraction $\frac{h\left(2x+h\right)}{h}$ by $h$
Evaluate the limit $\lim_{h\to0}\left(2x+h\right)$ by replacing all occurrences of $h$ by $0$
$x+0=x$, where $x$ is any expression
Access detailed step by step solutions to thousands of problems, growing every day!