3.9 Derivatives of Exponential and Logarithmic Functions

Exponential and logarithmic functions show up constantly in calculus because they model growth, decay, and many natural processes. Their derivatives follow clean, predictable rules, and once you know them, a huge range of problems becomes manageable.

Exponential and Logarithmic Differentiation

Differentiation of exponential functions

The core rule: the derivative of $a^x$ is $a^x \ln(a)$, where $a$ is a positive constant not equal to 1. You're multiplying the original function by the natural log of its base.

• $\frac{d}{dx}a^x = a^x \ln(a)$
• Example: $\frac{d}{dx}2^x = 2^x \ln(2)$

The natural exponential function $e^x$ is a special case. Since $\ln(e) = 1$, the derivative of $e^x$ is just $e^x$. This is the only function that is its own derivative, which is a big part of why $e$ is so important in calculus.

• $\frac{d}{dx}e^x = e^x$

Constant multiples work the way you'd expect:

• $\frac{d}{dx}(k \cdot a^x) = k \cdot a^x \ln(a)$
• $\frac{d}{dx}(5e^x) = 5e^x$

When the exponent is a function $u(x)$ rather than just $x$, apply the chain rule by multiplying by $u'(x)$:

• $\frac{d}{dx}e^{u(x)} = e^{u(x)} \cdot u'(x)$
• $\frac{d}{dx}a^{u(x)} = a^{u(x)} \cdot \ln(a) \cdot u'(x)$
• Example: $\frac{d}{dx}e^{\sin x} = e^{\sin x} \cdot \cos x$

Derivatives of logarithmic functions

The derivative of $\log_a(x)$ is $\frac{1}{x \ln(a)}$, where $a$ is a positive constant not equal to 1 and $x > 0$.

• $\frac{d}{dx}\log_a(x) = \frac{1}{x \ln(a)}$
• Example: $\frac{d}{dx}\log_2(x) = \frac{1}{x \ln(2)}$

For the natural logarithm, $\ln(a) = \ln(e) = 1$, so the formula simplifies:

• $\frac{d}{dx}\ln(x) = \frac{1}{x}$, where $x > 0$

Constant multiples carry through as usual:

• $\frac{d}{dx}(k \cdot \ln(x)) = \frac{k}{x}$
• Example: $\frac{d}{dx}(3\ln(x)) = \frac{3}{x}$

When the argument is a function $u(x)$, apply the chain rule. This gives you $u'(x)$ in the numerator:

• $\frac{d}{dx}\ln(u(x)) = \frac{u'(x)}{u(x)}$
• $\frac{d}{dx}\log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}$
• Example: $\frac{d}{dx}\ln(\cos x) = \frac{-\sin x}{\cos x} = -\tan x$

A common mistake: forgetting the chain rule when the argument isn't just $x$. If you see $\ln(\text{something})$, always ask yourself whether that "something" needs its own derivative.

Logarithmic differentiation techniques

Some functions are difficult or impossible to differentiate with standard rules alone. Logarithmic differentiation handles these by taking $\ln$ of both sides and then using implicit differentiation.

This technique is especially useful for:

• Functions of the form $[u(x)]^{v(x)}$, where both the base and exponent depend on $x$
• Products of many factors, like $f(x) = u_1(x) \cdot u_2(x) \cdot u_3(x)$

The key idea is that logarithm properties convert products into sums and exponents into coefficients, making differentiation much simpler.

Steps for logarithmic differentiation:

1. Set $y = f(x)$ and take the natural log of both sides: $\ln(y) = \ln(f(x))$
2. Use log properties to simplify: $\ln(ab) = \ln(a) + \ln(b)$ and $\ln(a^b) = b\ln(a)$
3. Differentiate both sides with respect to $x$. The left side becomes $\frac{1}{y} \cdot \frac{dy}{dx}$ by implicit differentiation
4. Solve for $\frac{dy}{dx}$ by multiplying both sides by $y$, then substitute the original expression back in for $y$

Example: Find $\frac{d}{dx}(x^x)$.

1. Let $y = x^x$, so $\ln(y) = \ln(x^x)$
2. Simplify: $\ln(y) = x \ln(x)$
3. Differentiate both sides: $\frac{1}{y} \cdot \frac{dy}{dx} = \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1$
4. Solve: $\frac{dy}{dx} = y(\ln(x) + 1) = x^x(\ln(x) + 1)$

Notice that in step 3, the right side uses the product rule on $x \ln(x)$. That's easy to miss if you rush through it.

Advanced Techniques and Applications

Two techniques from later in the course connect directly to exponential and logarithmic functions:

• The inverse function rule provides a way to derive the logarithmic derivative from the exponential derivative (and vice versa), since $\ln(x)$ and $e^x$ are inverses of each other.
• L'Hôpital's rule is used to evaluate limits that produce indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Many of these limits involve exponential or logarithmic expressions, so being comfortable with their derivatives is essential when you reach that topic.