Question

How do I find the following derivative?

Answer 1

\[m(t)=-2t(6t ^{4}-1)^{7}\]

Answer 2

I end up with the following, but it's incorrect:
\[-14t(6t ^{4}-1)^{6}(24t ^{3})\]

Answer 3

The correct answer is:
\[-2(6t ^{4}-1)^{6}(174t ^{4}-1)\]
I can't figure out how to get this answer

Answer 4

Use the product, power, and chain rules:$$-2t\times7(6t^4-1)^6\times24t^3+-2\times(6t^4-1)^7$$Notice our terms have \(-2(6t^4-1)^6\) in common, so factor it out:$$-2(6t^4-1)^6(t\times7\times24t^3+6t^4-1)$$Simplify the inner expression:$$(t\times7\times24t^3)+6t^4-1=168t^4+6t^4-1=174t^4-1$$... so our final derivative expression is just \(-2(6t^4-1)^6(174t^4-1)\)

Answer 5

Where does the -2x(6t^4-1)^7 come from?

Answer 6

I guess I'm having trouble applying the product, power, and chain rules.

Answer 7

@melbel the product rule; for our second term we have the product of the derivative of the first function (i.e. \(-2t\)) multiplied by the second function \(6t^4-1\)

Answer 8

The derivative involves the use of the Product Rule, Power Rule, and Chain Rule. First we use the product rule to get the following:\[m'(t)=(-2t)'(6t^4-1)^7+(-2t)[(6t^4-1)^7]'\] Now to differentiate the (6t^4-1) part, we need to use Power Rule:\[m'(t)=-2(6t^4-1)^7-2t[7(6t^4-1)^6]\] We are still not done because we must now apply the chain rule since (6t^4-1)^6 is a composite function so now we must multiply by the derivative of the stuff inside the brackets:\[m'(t)=-2(6t^4-1)^7-2t[7(6t^4-1)^6](24t^3)\] And that's it. We can factor and make it look nicer:\[m'(t)=-2[(6t^4-1)^7 +7t(6t^4-1)^6(24t^3)]=-2(6t^4-1)^6[(6t^4-1)+7t(24t^3)]\]
@melbel