Derivative of `e` to the Power of a Function: A Complete Explanation

📚 Understanding the Derivative of $e^{u(x)}$

The derivative of $e$ raised to a function, denoted as $e^{u(x)}$, is a common topic in calculus. Here, $u(x)$ represents any differentiable function of $x$. The formula to remember is:

$\frac{d}{dx} (e^{u(x)}) = e^{u(x)} \cdot \frac{du}{dx}$

In simpler terms, the derivative of $e^{u(x)}$ is the original function multiplied by the derivative of the exponent, $u(x)$. Let's explore the process with examples and then provide a complete explanation with a teacher's guide.

📝 Step-by-Step Explanation

• 🔍 Identify the Function: Recognize the function in the form $e^{u(x)}$. This means pinpointing what $u(x)$ actually is.
• 💡 Find the Derivative of the Exponent: Calculate $\frac{du}{dx}$. This step involves applying standard differentiation rules depending on what $u(x)$ is.
• ✏️ Apply the Formula: Use the formula $\frac{d}{dx} (e^{u(x)}) = e^{u(x)} \cdot \frac{du}{dx}$ to find the derivative.
• ✅ Simplify: Simplify the expression if possible.

🧪 Examples

Example 1: Find the derivative of $f(x) = e^{5x}$.

Here, $u(x) = 5x$, so $\frac{du}{dx} = 5$.

Applying the formula: $\frac{d}{dx} (e^{5x}) = e^{5x} \cdot 5 = 5e^{5x}$.

Example 2: Find the derivative of $f(x) = e^{x^2}$.

Here, $u(x) = x^2$, so $\frac{du}{dx} = 2x$.

Applying the formula: $\frac{d}{dx} (e^{x^2}) = e^{x^2} \cdot 2x = 2xe^{x^2}$.

Example 3: Find the derivative of $f(x) = e^{\sin(x)}$.

Here, $u(x) = \sin(x)$, so $\frac{du}{dx} = \cos(x)$.

Applying the formula: $\frac{d}{dx} (e^{\sin(x)}) = e^{\sin(x)} \cdot \cos(x) = \cos(x)e^{\sin(x)}$.

🧑‍🏫 Teacher's Guide: Derivative of $e^{u(x)}$

🎯 Objectives:

• 🎯 Students will be able to identify functions in the form $e^{u(x)}$.
• 🧠 Students will be able to find the derivative of $u(x)$.
• ✍️ Students will be able to apply the chain rule to find the derivative of $e^{u(x)}$.

🧰 Materials:

• 📝 Whiteboard or projector
• 🖊️ Markers or pens
• ➗ Calculus textbooks
• 💻 Worksheets with practice problems

Warm-up (5 mins):

• ⏰ Review basic differentiation rules (e.g., power rule, constant multiple rule).
• 💡 Quick quiz on finding derivatives of simple functions.

Main Instruction:

• 🗣️ Introduce the formula $\frac{d}{dx} (e^{u(x)}) = e^{u(x)} \cdot \frac{du}{dx}$.
• ✍️ Provide several examples (like the ones above), explaining each step clearly.
• 🤝 Encourage students to ask questions and work through examples together.
• ➕ Progress to more complex $u(x)$ functions (e.g., trigonometric, polynomial).

✅ Assessment:

• 📝 Provide a worksheet with practice problems of varying difficulty.
• ❓ Collect and grade the worksheets to assess understanding.
• 🧑‍🏫 Provide feedback and address any misconceptions.

🤔 Practice Quiz

Find the derivatives of the following functions:

1. ❓ $f(x) = e^{3x^2}$
2. ❓ $f(x) = e^{\cos(x)}$
3. ❓ $f(x) = e^{x^3 + 2x}$
4. ❓ $f(x) = e^{\tan(x)}$
5. ❓ $f(x) = e^{\sqrt{x}}$
6. ❓ $f(x) = e^{7x}$
7. ❓ $f(x) = e^{\ln(x)}$