How do you use the chain rule to differentiate #(e^(6x))^10#?

Question

Emma Aldridge · Accepted Answer

#60(e^(6x))^9*e^(6x)#

The chain rule is important when you are differentiating your equation. Make sure that you know the the different parts of your equation. Separate your equation into simply components.

#x^10# #e^x# #6x#

Remember that using the chain rule, you have to differentiate all of the components of the equation. When you do the chain rule, you have differentiate the first part and then keep the rest. then differentiate the second part and then keep the rest, then differentiate the last part.

The first component on the outside is #x^10# Therefore differentiate the first part and keep the rest #10(e^(6x))^9#

Then the next part is #e^x# keep the first part and Differentiate the second part and keep the rest #10(e^(6x))^9*e^(6x)#

The end is #6x# keep the first and second part, differentiate the last part #10(e^(6x))^9*e^(6x)*6#

Simplify and you will get #60(e^(6x))^9* e^(6x)#

Emily Ammerman · Answer

undefined To differentiate (e^(6x))^10 using the chain rule, first, we identify the inner function as e^(6x). Then, we raise it to the power of 10. The derivative of e^(6x) with respect to x is 6e^(6x). Applying the chain rule, we multiply this derivative by the exponent, which is 10. So, the final derivative is 10 * (e^(6x))^9 * 6e^(6x), which simplifies to 60 * e^(60x).