Find the derivative using the definition of the derivative for 𝑓(𝑥) = 2𝑥^3 + 5𝑥 − 3

Answer 1

Given: f(x) = 2𝑥³+ 5𝑥 − 3, find the derivative using definition of derivative.

The definition of derivative of f(x) denoted by f'(x) is:

f'(x) = lim_h→0 [f(x+h)-f(x)]/ h

𝑓(𝑥+h) = 2(𝑥+h)³ + 5(𝑥+h) − 3

= 2𝑥³+6x²h+6xh²+2h³ + 5𝑥+5h − 3

Plugin the value of f(x+h) and f(x) to the formula:

∴f '(x) = lim_h→0 [(2𝑥³+6x²h+6xh²+2h³ + 5𝑥+5h − 3)-(2𝑥³+ 5𝑥 − 3)]/h

Remove the parenthesis and combine like terms:

f '(x) = lim_h→0 (2𝑥³+6x²h+6xh²+2h³ + 5𝑥+5h − 3-2𝑥³ -5𝑥 + 3)/h

f '(x) = lim_h→0 (6x²h+6xh²+2h³ +5h)/h

Factor out h and reduce rational expression by canceling h:

f '(x) = lim_h→0 h(6x²+6xh+2h² +5)/h

f '(x) = lim_h→0 (6x²+6xh+2h² +5)

Plugin h = 0 to simplify limits:

f '(x) = 6x²+6x(0)+2(0)² +5

f '(x) = 6x² +5