How do you differentiate #f(x)= x * (4-x^2)^(1/2)# using the product rule?

1 Answer
Mar 5, 2016

Answer:

#f'(x) = (4 - 2x^2)/sqrt(4 - x^2)#

Explanation:

The Product Rule:

#frac{"d"}{"d"x}(uv) = v frac{"d"u}{"d"x} + u frac{"d"v}{"d"x}#

In this question,

  • #u = x#

  • #v = (4 - x^2)^{1/2}#

#f'(x) = frac{"d"}{"d"x}(x(4 - x^2)^{1/2})#

#= (4 - x^2)^{1/2} frac{"d"}{"d"x}(x) + x frac{"d"}{"d"x}((4 - x^2)^{1/2})#

#= (4 - x^2)^{1/2} (1) + x (1/2) (4 - x^2)^{-1/2} frac{"d"}{"d"x}(4 - x^2)#

#= sqrt(4 - x^2) + x (1/2) (4 - x^2)^{-1/2} (- 2x)#

#= sqrt(4 - x^2) - x^2/sqrt(4 - x^2)#

#= ((sqrt(4 - x^2))^2 - x^2)/sqrt(4 - x^2)#

#= (4 - 2x^2)/sqrt(4 - x^2)#