Overview
This lecture covers how to differentiate exponential functions with base (e), focusing on the chain rule when the exponent is a function of (x).
Derivatives of Exponential Functions
- The derivative of (e^x) with respect to (x) is (e^x).
- When the exponent is a function (u(x)), use the chain rule: (\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)).
- If the function has a coefficient, keep it with the derivative.
Example 1: (f(x) = e^{3x})
- Let (u = 3x), so (u' = 3).
- Derivative: (e^{3x} \cdot 3 = 3e^{3x}).
Example 2: (f(x) = 16e^{x^2})
- Let (u = x^2), so (u' = 2x).
- Derivative: (16e^{x^2} \cdot 2x = 32x e^{x^2}).
Example 3: (f(x) = 4e^{\sqrt{x}})
- Let (u = \sqrt{x}), so (u' = \frac{1}{2\sqrt{x}}).
- Derivative: (4e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} = \frac{2e^{\sqrt{x}}}{\sqrt{x}}).
Key Terms & Definitions
- Chain Rule — A differentiation rule used when a function is composed with another function.
- Exponential Function — A function of the form (e^{u(x)}), where (e) is the base of natural logarithms.
- Derivative — The rate at which a function changes with respect to its variable.
Action Items / Next Steps
- Practice differentiating more exponential functions with composite exponents.
- Review the application of the chain rule to other function types.