Q. 58 - Calculus

Question

Show that if $f$ is a function and $z=x+h$ then, 

              $\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$

Step-by-Step Solution

Answer

Ans:   The given expression $\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$ (It is true).

Step 1. Given information.

given,  

       $\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$

Step 2. Let f ( x ) be a function and z = x + h .

Then, 

   $z-x=h$

And

   $f\left(z\right)=f(x+h)$

The objective is to show that 

    $\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$

Replace $z=x+h$ and $z-x=h$ in $\frac{f\left(z\right)-f\left(x\right)}{z-x}$

$\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$

Hence, if $f$ is a function and $z=x+h$, then

$\frac{f\left(z\right)-f\left(x\right)}{z-x}=\frac{f(x+h)-f\left(x\right)}{h}$