Calculus I · 2A · lesson

Variable Bases and Exponents

Learn variable bases and exponents with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Course progress43 of 49 core pages43 of 49

Section overview

Implicit, inverse, and logarithmic differentiation

What this section is building

Learn variable bases and exponents with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Notice

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

Decide

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

Avoid

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

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 $x=1$ .

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Functions Such as

x^x

Concept

State the interval before taking logarithms

Explanation

Before the formulas

Explanation

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

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 $x=1$ .

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Functions Such as xxx^xxx

Related resources

Original instruction with traceable references.

Functions Such as xxx^xxx

Related resources

Original instruction with traceable references.

Functions Such as $x^x$

Functions Such as $x^x$