Question

Using the limit definition of a derivative, what is:5x^2 + 2x

Answer 1

Let:

`f(x)=5x^2+2x`

Using the limit definition of a derivative:

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

so:

`\begin{gathered} \lim _(h\to0)(5(x+h)^2+2(x+h)-5x^2-2x)/(h) \\ \lim _(h\to0)(5x^2+10xh+5h^2+2x+2h-5x^2-2x)/(h) \\ \end{gathered}`

Add like terms:

`\lim _(h\to0)(10xh+5h^2+2h)/(h)`

Divide the numerator and the denominator by h:

`\begin{gathered} \lim _(h\to0)10x+5h+2=10x+5(0)+2=10x+2 \\ so\colon \\ f^(\prime)(x)=10x+2 \end{gathered}`

Answer:

f'(x) = 10x + 2