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

##### 1 Answer

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

Then, using Product Rule,

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.