Calculate the derivative of the following function.

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Part 1

StartFraction dy Over dx EndFractiondydxequals=enter your response here

To calculate the derivative of the given function:

\[
y = \tan(e^x)
\]

We will use the chain rule, which states that:

\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]

Here, \( f(x) = \tan(x) \) and \( g(x) = e^x \).

### Step-by-step process:

1. **Differentiate the outer function (tangent):**
   The derivative of \( \tan(u) \) is \( \sec^2(u) \). So, applying this to \( \tan(e^x) \):

   \[
   \frac{d}{dx} \left( \tan(e^x) \right) = \sec^2(e^x)
   \]

2. **Differentiate the inner function (exponential):**
   The derivative of \( e^x \) is simply \( e^x \).

3. **Apply the chain rule:**
   Now, multiply the derivative of the outer function by the derivative of the inner function:

   \[
   \frac{dy}{dx} = \sec^2(e^x) \cdot e^x
   \]

Thus, the derivative of \( y = \tan(e^x) \) is:

\[
\frac{dy}{dx} = \sec^2(e^x) \cdot e^x
\]

Would you like further details or have any questions about this?

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Here are 5 related questions for further practice:

1. What is the derivative of \( y = \tan(2x) \)?
2. How do you differentiate \( y = \tan^{-1}(e^x) \)?
3. What is the second derivative of \( y = \tan(e^x) \)?
4. How does the chain rule apply to differentiating trigonometric functions?
5. What is the derivative of \( y = \sec(e^x) \)?

**Tip:** Always remember the chain rule when you have a function composed of multiple functions!

Calculate the derivative of the following function: y = tan(e^x).

Learn how to differentiate the function y = tan(e^x) step-by-step using the chain rule. This guide covers key calculus concepts, including the derivative of trigonometric functions and exponential functions. Perfect for students in Grades 11-12 and college-level calculus.

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