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 #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.