How do you differentiate #3sin^5(2x^2) # using the chain rule?

Question

How do you differentiate #3sin^5(2x^2)  # using the chain rule?

Cameron Alexander · Accepted Answer

Break the given chain rule problem in manageable links and use it to differentiate. Explanation is given below.

The Chain rule is easy if you split the chain into manageable links. Let us see how we can do this with respect to our problem #3sin^5(2x^2)# #y=3sin^5(2x^2)# We can split the links as below #y=3u^5# #u=sin(v)# #v=2x^# The chain rule is given by the following # dy/dx = dy/(du)* (du)/(dv) * (dv)/(dx)# Note how each link is differentiated separately and multiplied to form the chain. #y=3u^5# #dy/(du) = 15u^4# #u=sin(v)# #(du)/(dv) = cos(v)# #v=2x^2# #(dv)/(dx) = 4x# Applying chain rule we get #dy/dx = 15u^4*cos(v)*4x# #dy/dx = 60xu^4cos(v)# #dy/dx=60xsin^4(v)cos(v)# substituted #u=sin(v)# #dy/dx = 60xsin^4(2x^2)cos(2x^2)# substituted #v=2x^2# Final answer #dy/dx = 60xsin^4(2x^2)cos(2x^2)# To master try few problems and see how to split the links and differentiate. Practice would help you do similar problems with ease.

Ezekiel Alexander · Answer

undefined To differentiate $3\sin^5(2x^2)$ using the chain rule:

1. Identify the outer function: $u^5$, where $u = \sin(2x^2)$.
2. Identify the inner function: $v = 2x^2$.
3. Differentiate the outer function with respect to its variable:
   $\frac{d}{du}(u^5) = 5u^4$.
4. Differentiate the inner function with respect to its variable:
   $\frac{dv}{dx} = 4x$.
5. Combine the derivatives using the chain rule:
   $\frac{d}{dx}(3\sin^5(2x^2)) = 5(\sin(2x^2))^4 \cdot \frac{d}{dx}(\sin(2x^2))$.
6. Apply the derivative of the sine function:
   $\frac{d}{dx}(\sin(2x^2)) = \cos(2x^2) \cdot \frac{d}{dx}(2x^2)$.
7. Evaluate the derivative of the inner function:
   $\frac{d}{dx}(2x^2) = 4x$.
8. Substitute the derivatives back into the expression and simplify:
   $\frac{d}{dx}(3\sin^5(2x^2)) = 5(\sin(2x^2))^4 \cdot \cos(2x^2) \cdot 4x$.
9. Simplify further if needed.

That's the process for differentiating $3\sin^5(2x^2)$ using the chain rule.