differentiate : y=1/3e^5x^3 and y=2x2(x-1/3x^3)^2

Let's differentiate each function with respect to \( x \).

### 1. Differentiate \( y = \frac{1}{3}e^{5x^3} \):

To differentiate \( y = \frac{1}{3}e^{5x^3} \) with respect to \( x \), we can use the chain rule.

Given:
\[
y = \frac{1}{3}e^{5x^3}
\]

Let \( u = 5x^3 \), so \( y = \frac{1}{3}e^u \).

Now, differentiate \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{1}{3} \cdot e^u \cdot \frac{du}{dx}
\]

Differentiate \( u = 5x^3 \) with respect to \( x \):

\[
\frac{du}{dx} = 15x^2
\]

Substitute \( \frac{du}{dx} \) into the derivative:

\[
\frac{dy}{dx} = \frac{1}{3} \cdot e^{5x^3} \cdot 15x^2 = 5x^2 e^{5x^3}
\]

So, the derivative of \( y = \frac{1}{3}e^{5x^3} \) is:
\[
\frac{dy}{dx} = 5x^2 e^{5x^3}
\]

### 2. Differentiate \( y = 2x^2\left(x - \frac{1}{3}x^3\right)^2 \):

Here, we need to apply both the product rule and the chain rule.

Given:
\[
y = 2x^2\left(x - \frac{1}{3}x^3\right)^2
\]

Let's differentiate using the product rule:

\[
\frac{dy}{dx} = 2 \cdot \frac{d}{dx}\left[x^2\right] \cdot \left(x - \frac{1}{3}x^3\right)^2 + 2x^2 \cdot \frac{d}{dx}\left[\left(x - \frac{1}{3}x^3\right)^2\right]
\]

First, differentiate \( x^2 \):

\[
\frac{d}{dx}\left[x^2\right] = 2x
\]

Now, differentiate \( \left(x - \frac{1}{3}x^3\right)^2 \) using the chain rule:

Let \( v = x - \frac{1}{3}x^3 \), so \( y = 2x^2 v^2 \).

Differentiate \( v^2 \) with respect to \( v \):

\[
\frac{d}{dx}\left[v^2\right] = 2v \cdot \frac{dv}{dx}
\]

Differentiate \( v = x - \frac{1}{3}x^3 \) with respect to \( x \):

\[
\frac{dv}{dx} = 1 - x^2
\]

Substitute back:

\[
\frac{dy}{dx} = 2 \cdot 2x \cdot \left(x - \frac{1}{3}x^3\right)^2 + 2x^2 \cdot 2\left(x - \frac{1}{3}x^3\right) \cdot (1 - x^2)
\]

Simplify the expression:

\[
\frac{dy}{dx} = 4x\left(x - \frac{1}{3}x^3\right)^2 + 4x^2\left(x - \frac{1}{3}x^3\right)\left(1 - x^2\right)
\]

### Final Answers:
1. The derivative of \( y = \frac{1}{3}e^{5x^3} \) is:
\[
\frac{dy}{dx} = 5x^2 e^{5x^3}
\]

2. The derivative of \( y = 2x^2\left(x - \frac{1}{3}x^3\right)^2 \) is:
\[
\frac{dy}{dx} = 4x\left(x - \frac{1}{3}x^3\right)^2 + 4x^2\left(x - \frac{1}{3}x^3\right)\left(1 - x^2\right)
\]

Would you like a detailed breakdown of these steps, or do you have any questions?

Here are some related questions:
1. How does the chain rule work in differentiation?
2. What is the product rule in calculus?
3. Can you explain how to simplify the expression after differentiation?
4. How do you find the second derivative of these functions?
5. What are some practical applications of these derivatives?

**Tip:** When differentiating complex expressions, breaking down the problem using substitution and applying rules like the product and chain rule step by step can make the process more manageable.

Learn how to differentiate the functions y = 1/3e^5x^3 and y = 2x^2(x - 1/3x^3)^2 step-by-step. This problem involves applying the chain rule and product rule in calculus to find the derivatives. Suitable for advanced high school students and beyond.

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