Question

Find the derivative of 99x at x=100.

Answer 1

Hint: Here we will use the first principle derivative formula to find the derivative of 99x at x=100.

Complete step-by-step answer:
Let $f(x) = 99x$
The derivative of f(x) with respect to x is the function $f'(x)$ and is defined as
$f’(x) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f(x + h) - f(x)}}{h}$
Thus
$
  f'(100) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f(100 + h) - f(100)}}{h} \\
  {\text{ = }}\mathop {\lim }\limits_{h \to 0} \dfrac{{f(100 + h) - f(100)}}{h} \\
  {\text{ = }}\mathop {\lim }\limits_{h \to 0} \dfrac{{99(100 + h) - 99(100)}}{h}{\text{ [}}\because {\text{f(x) = 99x]}} \\
  {\text{ = }}\mathop {\lim }\limits_{h \to 0} \dfrac{{99 \times 100 + 99h - 99 \times 100}}{h} \\
  {\text{ = }}\mathop {\lim }\limits_{h \to 0} \dfrac{{99h}}{h} = 99 \\
$

Note: The derivative is a way to show the rate of change that is the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written using $\dfrac{{dy}}{{dx}}$ (meaning the difference in y divided by the difference in x). If we find the derivative of 99x wrt x normally, we will get 99 which is independent of x so we will get f’(100) = 99.