Here, we show you a step-by-step solved example of definition of derivative. This solution was automatically generated by our smart calculator:
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
Take the square of the first term: $x$
Twice ($2$) the product of the two terms: $x$ and $h$
Take the square of the second term: $h$
Add the three results, and we obtain the expanded polynomial
Expand the expression $\left(x+h\right)^2$ using the square of a binomial: $(a+b)^2=a^2+2ab+b^2$
Cancel like terms $x^{2}$ and $-x^2$
Expand the fraction $\frac{2xh+h^{2}}{h}$ into $2$ simpler fractions with common denominator $h$
Simplify the fraction $\frac{2xh}{h}$ by $h$
Simplify the fraction $\frac{h^{2}}{h}$ by $h$
Simplify the resulting fractions
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
Detaillierte Schritt-für-Schritt-Lösungen für Tausende von Problemen, die jeden Tag wachsen!