Calculus I · Unit 2A · lesson

Variable Bases and Exponents

Concept

Learning objectives

Differentiate expressions with a variable in both base and exponent.

Functions Such as $x^x$

Concept

State the interval before taking logarithms

The familiar real-valued formula for $f(x)^{g(x)}$ assumes $f(x)>0$ on the interval under discussion. Some variable powers extend to selected negative inputs, but they do not form one smooth real-valued rule across all negative bases. In Calculus I, state the positive-base interval explicitly rather than hiding a branch problem inside algebra.

Explanation

Before the formulas

In Variable Bases and Exponents, inverse and implicit ideas meet. Swapping input and output swaps horizontal and vertical change, so inverse slopes are reciprocals at corresponding points. Taking logarithms can also reveal hidden structure by turning products into sums and exponents into coefficients.

These methods are strategic transformations, not new definitions of derivative. State the domain assumptions, preserve the original relationship, and substitute back at the end. A clean solution explains why the transformation helps before carrying out the algebra.

Explanation

When both the base and exponent vary, neither the ordinary power rule nor the ordinary exponential rule is enough

The function $x^x$ changes in two ways: the base changes and the exponent changes. Logarithmic differentiation handles both at once by rewriting $\ln y=x\ln x$ , where the product rule can see the two contributions.

The same method applies to $f(x)^{g(x)}$ on intervals where the logarithm is defined. Domain assumptions should be stated rather than smuggled past the reader.

The power rule handles a variable base with constant exponent, while the exponential rule handles a constant base with variable exponent. A function such as $x^x$ changes in both places, so neither rule alone applies.

Logarithmic differentiation separates those roles. The resulting derivative contains both $\ln x$ , which records variation in the exponent, and a constant term, which records variation in the base.

Neither the power rule nor the ordinary exponential rule applies directly to $x^x$ . Logarithms convert the exponent into a product.

Guided walkthrough

Differentiate $x^x$

Let

y=x^x,\qquad x>0.

Take natural logs:

\ln y=x\ln x.

Differentiate:

\frac{y'}y=\ln x+1.

Therefore

\boxed{y'=x^x(\ln x+1)}.

Worked example

A function raised to a function

For $y=[g(x)]^{h(x)}$ with $g(x)>0$ ,

\ln y=h(x)\ln g(x),

\boxed{y'=g(x)^{h(x)}\left[h'(x)\ln g(x)+h(x)\frac{g'(x)}{g(x)}\right]}.

Exercise

Differentiate $x^{\sin x}$ for $x>0$ .

Exercise

Differentiate $(x^2+1)^x$ .

Exercise

Differentiate $(\sin x)^x$ on an interval where $\sin x>0$ .

Exercise

Find the slope of $y=x^x$ at $x=1$ .

After the explanation

Use the section idea

Reading lens

Track which variable depends on which and use reciprocal or logarithmic structure only where its conditions hold.

Mental model

Implicit equations constrain variables together; inverse functions exchange inputs and outputs; logarithms turn products and powers into sums.

Decision

Choose implicit, inverse, or logarithmic differentiation from the equation's representation, not from surface complexity.

Common trap

Dropping a y-prime factor, using a reciprocal slope at the wrong point, or ignoring domain restrictions.

Check yourself

Can you identify the correspondence point and all hidden dependencies before differentiating?

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Functions Such as xxx^xxx

Original instruction with traceable references.

Functions Such as $x^x$