Find the derivative of f(y)=4sin^9(y^6) **Understanding and Finding the Derivative of a Trigonometric Function**The problem provided asks us to find the derivative of \( f(y) = 4\sin^9\left( y^6 \right) \).### Instructions:1. **Enclose arguments of functions in parentheses.** - For example, when you see \(\sin(2x)\), it indicates the sine of \(2x\). - Include a multiplication sign between symbols, so input this as \(\sin(2*x)\) when entering it manually.2. **Writing the Power of a Trigonometric Function:** - When expressing the power of a trigonometric function, remember that writing \(\sin^4(x)\) is just a shorthand method. - To be mathematically correct and align with systems such as Mobius, you should write this as \((\sin(x))^4\).### Solution:To find the derivative \( f'(y) \), you will apply the chain rule.### Explanation of Derivative Calculation:1. **Given Function:** \[ f(y) = 4\sin^9(y^6) \] 2. **Applying the Chain Rule:** - Let \(u = y^6\) - Hence, \(f(y) = 4\sin^9(u)\) - Then, \(f(y) = 4\left(\sin(u)\right)^9\)3. **Differentiate with respect to \(u\):** \[ \frac{d}{du}\left[4(\sin(u))^9\right] = 4 \cdot 9 (\sin(u))^8 \cdot \cos(u) \] - We have used the chain rule here, differentiating \((\sin(u))^9\) with respect to \(u\).4. **Substitute back \(u = y^6\):** \[ f'(y) = 36(\sin(y^6))^8 \cdot \cos(y^6) \cdot \frac{d}{dy}(y^6) \]5. **Differentiate \(y^6\):** \[ \frac{d}{dy}(y^6) = 6y^5 \]6. **Combine the results:**

Name: Find the derivative of f(y)=4sin^9(y^6) **Understanding and Finding th
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: Find the derivative of f(y)=4sin^9(y^6) **Understanding and Finding the Derivative of a Trigonometric Function**The problem provided asks us to find the derivative of \( f(y) = 4\sin^9\left( y^6 \right) \).### Instructions:1. **Enclose arguments of f