Question

Can anyone explain (step-by-step) how to work out this calculus problem?

Answer 1

Step-by-step explanation:

Use the difference quotient

$\frac{f(x + h) - f(x)}{h}$

Here it will become

$\frac{2 (0.85) {}^{x + h} - 2(0.85 {}^{x}) }{h}$

For the first box, let x=-4, and h= 0.001

$\frac{2(0.85) {}^{ - 4 + 0.001} - 2(0.85) {}^{ - 4} }{0.001}$

Using a calculator, you will get about

$- 0.622$

Next, let do this for 0.5

$\frac{2(0.85) {}^{0.5 + 0.001} - 2(0.85) {}^{0.5} }{0.001}$

You will get about

$- 0.3$

Finally, do this for

1.25

$\frac{2(0.85) {}^{1.25 + 0.001} - 2(0.85) {}^{1.25} }{0.001}$

You get about

$- 0.265$

Note : This is the definition of a derivative