Exponential Function Differentiation

Exponential Function Differentiation

When differentiating exponential functions, the key idea is that the derivative keeps the same exponential form, but may gain a constant factor from the chain rule.

1) Identify the exponential part

Look for expressions like:

• (e^{u(x)})
• (a^{u(x)}), where (a>0) and (a\neq 1)

The exponent may be just (x), or it may be a more complicated function.

2) Use the correct rule

• For (e^{u(x)}): [ \frac{d}{dx}\big(e^{u(x)}\big)=u'(x)e^{u(x)} ]
• For (a^{u(x)}): [ \frac{d}{dx}\big(a^{u(x)}\big)=u'(x)a^{u(x)}\ln(a) ]

If the exponent is simply (x), then (u'(x)=1), so the result is especially simple.

3) Simplify the final expression

After differentiating, combine constants and write the answer in a clean form. If there are multiple factors, multiply them carefully and reduce where possible.

4) Check your work

A good check is to see whether the derivative still contains the original exponential function and whether any chain-rule factor from the exponent is included. If the exponent was linear, the derivative should be the original exponential times a constant.

Example pattern

If (f(x)=e^{3x-1}), then the exponent derivative is (3), so [ f'(x)=3e^{3x-1}. ]

Use this same method for each problem: identify the base, differentiate the exponent, apply the rule, and simplify.