Proof of Derivative of \( e^x \)
The proof of the derivative of the natural exponential \( e^x \) is presented using the limit definition of the derivative. The derivative of a composite function of the form \( e^{u(x)} \) is also presented including examples with their solutions.
Proof of the Derivative of \( e^x \) Using the Definition of the Derivative
The definition of the derivative \( f' \) of a function \( f \) is given by the limit
\[ f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h} \]
Let \( f(x) = e^x \) and write the derivative of \( e^x\) as follows
Derivative of the Composite Function \( y = e^{u(x)} \)We now consider the composite exponential of another function u(x). Use the chain rule of differentiation to write\( \displaystyle \dfrac{d}{dx} e^{u(x)} = \dfrac{d}{du} e^{u(x)} \dfrac{d}{dx} u \) Simplify \( = e^u \dfrac{d}{dx} u \) Conclusion \[ \displaystyle \dfrac{d}{dx} e^{u(x)} = e^u \dfrac{d}{dx} u \]
Example 1
Solution to Example 1
More References and linksderivativedefinition of the derivative Chain Rule of Differentiation in Calculus. |