Question

How do you use the product rule to find the derivative of `y=(1/x^2-3/x^4)*(x+5x^3)` ?

Answer 1

The Answer is

`y'=(1/x^2-3/x^4)*(1+15x^2)+(-2/x^3+12/x^5)(x+5x^3)`

Solution :
Suppose we have `y=f(x)*g(x)`
Then, using Product Rule, `y'=f(x)*g'(x)+f'(x)*g(x)`

In simple language, keep first term as it is and differentiate the second term, then differentiate the first term and keep the second term as it is or vice-versa.

So, here if we consider,

`f(x)=(1/x^2-3/x^4)`
`g(x)=(x+5x^3)`

Then,

`f'(x)=(-2/x^3+12/x^5)`
`g'(x)=(1+15x^2)`

Hence, using the product rule,

`y'=(1/x^2-3/x^4)(1+15x^2)+(-2/x^3+12/x^5)(x+5x^3)`

In case , if we have more than two function, let see

`y=u(x)*v(x)*w(x)`

then,

`y'=u'(x)*v(x)*w(x)+u(x)*v'(x)*w(x)+u(x)*v(x)*w'(x)`

i.e. differentiate one function at a time and keep the remaining two as it is or consider them as constant and similarly follow for the remaining two.