cos(x)^3*sin(x) calculate derivative wrt x 2 times

To find the second derivative of \( \cos^3(x) \sin(x) \) with respect to \( x \), let's proceed step by step.

### Step 1: First Derivative
Given function:
\[
f(x) = \cos^3(x) \sin(x)
\]

Using the product rule:
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
where \( u(x) = \cos^3(x) \) and \( v(x) = \sin(x) \).

First, calculate the derivatives of \( u(x) \) and \( v(x) \):
\[
u(x) = \cos^3(x) \implies u'(x) = 3 \cos^2(x) \cdot (-\sin(x)) = -3 \cos^2(x) \sin(x)
\]
\[
v(x) = \sin(x) \implies v'(x) = \cos(x)
\]

Now apply the product rule:
\[
\frac{d}{dx} \left[\cos^3(x) \sin(x)\right] = -3 \cos^2(x) \sin(x) \cdot \sin(x) + \cos^3(x) \cdot \cos(x)
\]
Simplify:
\[
f'(x) = -3 \cos^2(x) \sin^2(x) + \cos^4(x)
\]

### Step 2: Second Derivative
Now, take the derivative of \( f'(x) \) to find \( f''(x) \).

Differentiate each term separately:
\[
\frac{d}{dx} \left[-3 \cos^2(x) \sin^2(x)\right]
\]
Use the product rule:
\[
\text{Let } g(x) = -3 \cos^2(x), \, h(x) = \sin^2(x)
\]
\[
g'(x) = -6 \cos(x) \cdot (-\sin(x)) = 6 \cos(x) \sin(x)
\]
\[
h'(x) = 2 \sin(x) \cos(x)
\]
Then:
\[
\frac{d}{dx} \left[-3 \cos^2(x) \sin^2(x)\right] = 6 \cos(x) \sin(x) \cdot \sin^2(x) - 3 \cos^2(x) \cdot 2 \sin(x) \cos(x)
\]
Simplify:
\[
= 6 \cos(x) \sin^3(x) - 6 \cos^3(x) \sin(x)
\]

Next, differentiate the second term \( \cos^4(x) \):
\[
\frac{d}{dx} \left[\cos^4(x)\right] = 4 \cos^3(x) \cdot (-\sin(x)) = -4 \cos^3(x) \sin(x)
\]

Thus, the second derivative is:
\[
f''(x) = 6 \cos(x) \sin^3(x) - 6 \cos^3(x) \sin(x) - 4 \cos^3(x) \sin(x)
\]
Simplify further:
\[
f''(x) = 6 \cos(x) \sin^3(x) - 10 \cos^3(x) \sin(x)
\]

### Final Answer:
\[
f''(x) = 6 \cos(x) \sin^3(x) - 10 \cos^3(x) \sin(x)
\]

Would you like more details on this process or have any other questions? 

Here are 5 related questions:
1. How can you simplify trigonometric expressions involving powers of sine and cosine?
2. What are the common rules for finding derivatives of trigonometric functions?
3. How do you apply the product rule in the context of higher derivatives?
4. Can you calculate the derivative of a more complex trigonometric function?
5. How does the chain rule apply in differentiating nested trigonometric functions?

**Tip:** When differentiating trigonometric functions, always remember to apply the chain rule correctly, especially when the function is raised to a power.

Learn how to calculate the second derivative of cos^3(x) * sin(x) step-by-step. Understand the application of the product rule and differentiation of trigonometric functions. Suitable for college-level students and calculus enthusiasts.

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