Question

What is the derivative of `(5^xroot4(x^3))/(x^7(x+1)^2)`?

Answer 1

`y'=(5^x*x^(3/4))/(x^7*(x+1)^2)(ln(5)+3/(4x)-7/x-2/(1+x))`

Explanation:

One could use the rules of derivate very systematically

Another approach is to use the properties of logarithms,
which often is easier when there are multiple functions in the numerator and denominator

We have the function

`y=(5^x*x^(3/4))/(x^7*(x+1)^2)`

Take the logarithm on both sides

`ln(y)=ln((5^x*x^(3/4))/(x^7*(x+1)^2))`

Apply the rules of logarithms on the RHS

`ln(y)=ln(5^x*x^(3/4))-ln(x^7*(x+1)^2)`

`ln(y)=xln(5)+3/4ln(x)-7ln(x)-2ln(x+1)`

Differentiate both sides (notice the implicit differentiation on LHS)

`y'1/y=ln(5)+3/(4x)-7/x-2/(1+x)`

`y'=y(ln(5)+3/(4x)-7/x-2/(1+x))`

Substitute `y=(5^x*x^(3/4))/(x^7*(x+1)^2)`

`y'=(5^x*x^(3/4))/(x^7*(x+1)^2)(ln(5)+3/(4x)-7/x-2/(1+x))`