Final Answer
Step-by-step Solution
Specify the solving method
Find the derivative of $\frac{\left(x-1\right)^3}{3}$ 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 $\frac{\left(x-1\right)^3}{3}$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
Learn how to solve definition of derivative problems step by step online.
$\lim_{h\to0}\left(\frac{\frac{\left(x+h-1\right)^3}{3}-\frac{\left(x-1\right)^3}{3}}{h}\right)$
Learn how to solve definition of derivative problems step by step online. Find the derivative of ((x-1)^3)/3 using the definition. Find the derivative of \frac{\left(x-1\right)^3}{3} 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 \frac{\left(x-1\right)^3}{3}. Substituting f(x+h) and f(x) on the limit, we get. Combine \frac{\left(x+h-1\right)^3}{3}-\frac{\left(x-1\right)^3}{3} in a single fraction. Divide fractions \frac{\frac{\left(x+h-1\right)^3-\left(x-1\right)^3}{3}}{h} with Keep, Change, Flip: \frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}. Factor the sum or difference of cubes using the formula: a^3\pm b^3 = (a\pm b)(a^2\mp ab+b^2).