How do you differentiate y=(2x-5)^4(8x^2-5)^-3?

1 Answer
Jan 23, 2017

y'=(8(2x-5)^3(-4x^2+30x-5))/((8x^2-5)^4)

Explanation:

You would apply the rule:

y=f(x)g(x)->y'=color(red)(f'(x))g(x)+f(x)color(green)(g'(x))

and the chain rule, since

f(x)=(2x-5)^4 and g(x)=(8x^2-5)^-3

Then:

y'=color(red)(4(2x-5)^3*2)(8x^2-5)^-3+(2x-5)^4*color(green)((-3)(8x^2-5)^-4*16x)

=8(2x-5)^3/(8x^2-5)^3-48x(2x-5)^4/(8x^2-5)^4

=(8(8x^2-5)(2x-5)^3-48x(2x-5)^4)/((8x^2-5)^4)

=(8(2x-5)^3(8x^2-5-12x^2+30x))/((8x^2-5)^4)

=(8(2x-5)^3(-4x^2+30x-5))/((8x^2-5)^4)