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

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Answer 1 · 2018-03-13T21:15:22

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

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

#dy/dx [ (x^3 + 2)^2(x^5 + 4)^4]#

#=d/dx[(x^3+2)^2]⋅(x^5+4)^4+(x^3+2)2⋅d/dx[(x^5+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(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(3x^2+0)(x^3+2)(x^5+4)^4+4(5x^4+0)(x^3+2)^2(x^5+4)^3#

#=6x^2(x^3+2)(x^5+4)^4+20x^4(x^3+2)^2(x^5+4)^3#

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