Q. 3 - Calculus

Question

Use limits to give mathematical definitions for each of the following derivatives, first with the $h\to 0$ definition of the derivative, and then with the $z\to x$definition:

(a) the derivative of a function f at the point $x=5.$

(b) the derivative of a function f

(c) the right derivative of a function f at the point $x=-2.$

Step-by-Step Solution

Answer

The solution are:$f^{\text{'}}\left(5\right)=\underset{h\to 0}{\lim }\frac{f\left(5+h\right)-f\left(5\right)}{h}$

                            :$f^{\text{'}}\left(x\right)=\underset{z\to x}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}$

                            :$f^{\text{'}}(-2)=\underset{h\to 0}{\lim }\frac{f\left(-2+h\right)-f\left(-2\right)}{h}$

Part(a) Step1.Given Information

Given equation the derivative at $x=5.$

Step2.The solution

The solution is $$\begin{gathered} f^{\text{'}}\left(5\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ {f}^{\text{'}}\left(5\right)=\underset{h\to 0}{\lim }\frac{f\left(5+h\right)-f\left(5\right)}{h}. \end{gathered}$$

Part(b) Step1.Given Information

Given equation of function $f^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Step2.The solution

the solution is $\underset{z\to x}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}.$

Part(c) Step1.Given Information

Given equation is at $x=-2.$

Step2.The solution

The solution is $$\begin{gathered} =\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{f\left(-2+h\right)-f\left(-2\right)}{h} . \end{gathered}$$