Question

Find the derivative of the following function, using the various rules for differentiation

Answer 1

Simplifying
`f(x)` will let you avoid using the quotient rule:


`f(x)=(2x^4+1)/(7x^4)+3\sqrt[3]{x^2}-4x^(-1/4)`

`f(x)=\frac27+\frac17x^(-4)+3x^(2/3)-4x^(-1/4)`

and now you're just taking derivatives of power functions. You should get


`f'(x)=0-\frac47x^(-5)+2x^(-1/3)+x^(-5/4)`

and we can rewrite this as


`f'(x)=-\frac4{7x^5}+\frac2{\sqrt[3]{x}}+\frac1{x^(5/4)}`

even combining into one fraction as


`f'(x)=-\frac17x^(-5)\left(4-14x^(14/3)-7x^(15/4)\right)=-(4-14x^(14/3)-7x^(15/4))/(7x^5)`