How do you use the chain rule to differentiate #y=(4x^3-7)^4(3x+2)^10#?

I got:

#6(4x^3 - 7)^3(3x + 2)^9(44x^3 + 16x^2 - 35)#

If you can't validly modify your answer to match this one, you've mis-factored somewhere.

You would use the product rule on these terms, and the chain rule while using the product rule.

Product Rule:

#\mathbf(d/(dx)[f(x)g(x)] = f(x)g"'"(x) + g(x)f"'"(x))#

Chain Rule:

#\mathbf(d/(dx)[f(g(x))] = f"'"(g(x))*g"'"(x))#

Therefore, what you have (without the chain rule) is:

#d/(dx)[(4x^3 - 7)^4(3x + 2)^10]#

#= (4x^3 - 7)^4*10(3x + 2)^9 + (3x + 2)^10*4(4x^3 - 7)^3#

Now, incorporate the chain rule to finish this step and get:

#= (4x^3 - 7)^4*10(3x + 2)^9*3 + (3x + 2)^10*4(4x^3 - 7)^3*12x^2#

Now, group terms before factoring:

#= 30(4x^3 - 7)^4(3x + 2)^9 + 48x^2(4x^3 - 7)^3(3x + 2)^10#

Factor out common terms:

#= (4x^3 - 7)^3[30(4x^3 - 7)(3x + 2)^9 + 48x^2(3x + 2)^10]#

#= (4x^3 - 7)^3(3x + 2)^9[30(4x^3 - 7) + 48x^2(3x + 2)]#

Factor a little bit more:

#= 6(4x^3 - 7)^3(3x + 2)^9[5(4x^3 - 7) + 8x^2(3x + 2)]#

And maybe distribute inner terms:

#= 6(4x^3 - 7)^3(3x + 2)^9(20x^3 - 35 + 24x^3 + 16x^2)#

#= color(blue)(6(4x^3 - 7)^3(3x + 2)^9(44x^3 + 16x^2 - 35))#

That's probably simplified enough. Either way, it's correct.