Question

Use the definition of the derivative to show f’(x)=2x+2 if f(x)=x^2+2x this means the long way by substituting into the limit: f’ (x)=lim h 0 f(x+h)-f(x)/h

Answer 1

Given: A function f(x)

`f(x)=x^2+2x`

Required: To verify the derivative of the given function is

`f^(\prime)(x)=2x+2`

by using the method of Limits.

Step-by-step explanation: The derivative of a function can be calculated by using limits-

`f^(\prime)(x)=\lim_(h\to0)(f(x+h)-f(x))/(h)`

Here we have

`\begin{gathered} f(x)=x^2+2x \\ f(x+h)=(x+h)^2+2(x+h) \\ =x^2+h^2+2xh+2x+2h \end{gathered}`

Putting these values in Limit we get

`\begin{gathered} f^(\prime)(x)=\lim_(h\to0)(x^2+h^2+2xh+2x+2h-x^2-2x)/(h) \\ =\lim_(h\to0)(h^2+2xh+2h)/(h) \\ =\lim_(h\to0)(h(h+2x+2))/(h) \\ =2x+2 \end{gathered}`

Hence the result is verified.

Final Answer: The derivative of the given function is

`f^(\prime)(x)=2x+2`