How do you differentiate # f(x)=(3x^5 - 4x^3 + 2)^23 # using the chain rule.?

1 Answer
Dec 9, 2015

Answer:

#f'(x)= 69x^2(3x^5 -4x^3+2)^22 (5x^2 -4)#

Explanation:

Remember: Chain rule:

#"Derivative of " f(g(x))= f'(x)g(x)*g'(x)#

Derivative of Power and chain rule: #f(x)= (g(x))^n = f'(x) = n(g(x)^(n-1))*g'(x) #

Given #f(x) (3x^5 -4x^3+2)^23#
#f'(x) = 23(3x^5-4x^3+2)^(23-1) * color(red)(d/(dx)(3x^5 -4x^3+2)#

#=23(3x^5 -4x^3+2)^22 color(red)((15x^4 -12x^2+0) #
#=23(3x^5 -4x^3+2)^22color(red) (15x^4 -12x^2)# or

by factor out the greatest common factor #color(blue)(3x^2)#from #15x^4 -12x^2#
#f'(x) = 23*color(blue)(3x^2)(3x^5 -4x^3+2)^22 (5x^2 -4)#

Simplify:
#f'(x)= 69x^2(3x^5 -4x^3+2)^22 (5x^2 -4)#