Question

The given limit represents the derivative of a function f at a number a Find f and .a

$\underset{h\to 0}{\lim }\frac{{(1+h)}^{10}-1}{h}$

Short answer

The value of f is$x^{10}$and value ofis 1.

Step-by-step solution

Step 1. Write the given expression.

The expression is$\underset{h\to 0}{\lim }\frac{{(1+h)}^{10}-1}{h}$ .

Step 2. Write the definition of derivative.

The derivative of$f\left(a\right)$with respect tois the function$f\text{'}\left(a\right)$and is defined as:$f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}\mathrm{........}\left(i\right)$

Step 3. Find the value of f and a.

Compare equation (i) with the given expression. So, we get

$\begin{array}{l}f\left(x\right)=x^{10}\\ f\left(a\right)=1^{10}\\ a=1\end{array}$

Therefore, the value of f is$x^{10}$ and value of is1 .