Question

How do you use the definition of a derivative to find the derivative of `f(x)=x^3+5x^2+6`?

Answer 1

`(df(x))/(dx)=3x^2+10x`

Explanation:

As per definition of a derivative of function `f(x)`,

`(df(x))/(dx)=Lt_(h->0)(f(x+h)-f(x))/h`

Here `f(x)=x^3+5x^2+6` and hence

`f(x+h)=(x+h)^3+5(x+h)^2+6` and

`f(x+h)-f(x)=(x+h)^3+5(x+h)^2ul(+6)-x^3-5x^2ul(-6)`

= `x^3+3x^2h+3xh^2+h^3+5(x^2+2hx+h^2)-x^3-5x^2`

= `ul(x^3)+3x^2h+3xh^2+h^3+ul(5x^2)+10hx+5h^2ul(-x^3-5x^2)`

= `3x^2h+3xh^2+h^3+10hx+5h^2` and

`(df(x))/(dx)=Lt_(h->0)(3x^2h+3xh^2+h^3+10hx+5h^2)/h`

= `Lt_(h->0)3x^2+3xh+h^2+10x+5h`

= `3x^2+10x`