Question

Using the limit definition, how do you differentiate `f(x)=3x^5 + 4x`?

Answer 1

`f'(x)=15x^4+4`

Explanation:

For theegiven function:
`f(x)=3x^5+4x`

We take the limit definition for a derivative:
`f'(x)=lim_(h rarr 0)(f(x+h)-f(x))/h`

This gives:
`f'(x)=lim_(h rarr 0)((3(x+h)^5+4(x+h))-(3x^5+4x))/h`

`=lim_(h rarr 0)(3(cancel(x^5)+5x^4h+10x^3h^2+10x^2h^3+5xh^4+h^5)+cancel(4x)-cancel(3x^5)-cancel(4x)+4h)/h`

`=lim_(h rarr 0)3(5x^4+10x^3h+10x^2h^2+5xh^3+h^4)+4`

Taking the limit as `h` tends to zero on this expression gives us:
`f'(x)=15x^4+4`