How do you differentiate y = (x^3 + 2)^2(x^5 + 4)^4y=(x3+2)2(x5+4)4?

1 Answer
Mar 13, 2018

To differentiate the expression, you could you explicit differentiation or implicit differentiation. I'm going to use implicit differentiation.

Explanation:

Using the chain rule in conjunction with the product rule, we get:

dy/dx [ (x^3 + 2)^2(x^5 + 4)^4]dydx[(x3+2)2(x5+4)4]

=d/dx[(x^3+2)^2]⋅(x^5+4)^4+(x^3+2)2⋅d/dx[(x^5+4)^4]=ddx[(x3+2)2](x5+4)4+(x3+2)2ddx[(x5+4)4]

=2(x^3+2)⋅d/dx[x^3+2]⋅(x^5+4)^4+4(x^5+4)^3⋅d/dx[x^5+4]⋅(x^3+2)^2=2(x3+2)ddx[x3+2](x5+4)4+4(x5+4)3ddx[x5+4](x3+2)2

=2(d/dx[x^3]+d/dx[2])(x^3+2)(x^5+4)^4+4(d/dx[x^5]+d/dx[4])(x3+2)^2(x^5+4)^3=2(ddx[x3]+ddx[2])(x3+2)(x5+4)4+4(ddx[x5]+ddx[4])(x3+2)2(x5+4)3

=2(3x^2+0)(x^3+2)(x^5+4)^4+4(5x^4+0)(x^3+2)^2(x^5+4)^3=2(3x2+0)(x3+2)(x5+4)4+4(5x4+0)(x3+2)2(x5+4)3

=6x^2(x^3+2)(x^5+4)^4+20x^4(x^3+2)^2(x^5+4)^3=6x2(x3+2)(x5+4)4+20x4(x3+2)2(x5+4)3

Which, when simplified, is equal to:
=2x^2(x^3+2)(x^5+4)^3(13^x5+20x^2+12)=2x2(x3+2)(x5+4)3(13x5+20x2+12)