use the definition of derivative to show that d/dx f(x)=-2x

Question

let f(x)= 3-x²

Answer 1

We will use the limit definition of the derivative function to show this:

f^'(x) = lim_h→0 [f(x+h) - f(x)] / h (This is a useful definition to memorize, btw.)

f(x) = 3 - x²

f^'(x) = lim_h→0 [(3 - (x+h)² - (3 - x²)] / h

= lim_h→0 [ 3 - x² - 2xh - h² - 3 + x²] / h

= lim_h→0 [ - 2xh - h² ] / h

= lim_h→0 - 2x - h

= -2x

Notice that using this limit definition, we should always get the terms without an h in the numerator to cancel, leaving only terms with an h there. Then we can factor out h, cancel with the denominator, and then let h = 0 to determine the limit.