Derivative of Logarithmic Functions: A Hard Task Made Easy
TL;DR: The natural log has the friendliest derivative in calculus: the derivative of ln x is just 1 over x. For a log with a different base, you tack on a ln of that base in the denominator. When ln wraps a function, the chain rule kicks in, the derivative becomes u-prime over u. And for products and powers that look like a mess, there is a slick move called logarithmic differentiation that turns them into manageable sums.
Key takeaways:
- \((\ln x)' = 1/x\).
- \((\log_a x)' = 1/(x \ln a)\).
- Chain rule: \((\ln u)' = u'/u\).
- Logarithmic differentiation handles tough products, quotients, and \(f(x)^{g(x)}\) forms.
- Domain matters: \(\ln x\) is defined only for \(x > 0\).
Derivatives of logarithmic functions involve understanding how the logarithm's rate of change relates to its base and argument. For the natural logarithm \( ln(x) \), the derivative is \( \frac{1}{x} \). When dealing with more complex logarithmic expressions, the chain rule is often employed, especially if the logarithm's argument is a function itself, not just a simple variable.
The formulas to find the derivative of logarithmic functions:
Logarithm of \( x \) to the base of \( a \):
The derivative of \( \log_a x \): \( \left(\log_a x\right)’ = \frac{1}{x \ln a} \)
Example:
\( \text{Given function: } \log_2 x \)
\( \text{Derivative: } \left(\log_2 x\right)’ = \frac{1}{x \ln 2} \)
Logarithm of \( f(x) \) to the base of \( a \):
The derivative of \( \log_a f(x) \): \( \left(\log_a f(x)\right)’ = \frac{f'(x)}{f(x) \ln a} \)
Example:
\( \text{Given function: } \log_3 (x^2 + 1) \)
\( \text{Derivative: } \left(\log_3 (x^2 + 1)\right)’ = \frac{2x}{(x^2 + 1) \ln 3} \)
Natural logarithm of \( x \):
The derivative of \( \ln x \): \( \left(\ln x\right)’ = \frac{1}{x} \)
Example:
\( \text{Given function: } \ln (3x) \)
\( \text{Derivative: } \left(\ln (3x)\right)’ = \frac{1}{3x} \cdot 3 = \frac{1}{x} \)
Natural logarithm of \( f(X) \):
The derivative of \( \ln f(x) \): \( \left(\ln f(x)\right)’ = \frac{f'(x)}{f(x)} \)
Example:
\( \text{Given function: } \ln (x^3 + 2x) \)
\( \text{Derivative: } \left(\ln (x^3 + 2x)\right)’ = \frac{3x^2 + 2}{x^3 + 2x} \)
Examples:
Let’s consider a complex example:
\( \text{Find the derivative of } h(x) = \ln(x) \cdot \log_2(x^2 + 1) \)
1. Apply the product rule
\( h'(x) = f'(x)g(x) + f(x)g'(x) \)
2. Define \( f'(x) \) and \( g'(x) \)
\( f(x) = \ln(x) \rightarrow f'(x) = \frac{1}{x} \)
\( g(x) = \log_2(x^2 + 1) \rightarrow g'(x) = \frac{2x}{(x^2 + 1) \ln 2} \)
3. Combine using the product rule
\( h'(x) = \frac{1}{x} \cdot \log_2(x^2 + 1) + \ln(x) \cdot \frac{2x}{(x^2 + 1) \ln 2} \)
Here is another example involving radicals:
\( \text{Find the derivative of } h(x) = \sqrt{\ln x} \cdot \log_2(x^3 + 1) \)
1. Apply the product rule
\( h'(x) = f'(x)g(x) + f(x)g'(x) \)
2. Define \( f'(x) \) and \( g'(x) \)
\( f(x) = \sqrt{\ln x} \rightarrow f'(x) = \frac{1}{2\sqrt{\ln x}} \cdot \frac{1}{x} \)
\( g(x) = \log_2(x^3 + 1) \rightarrow g'(x) = \frac{3x^2}{(x^3 + 1) \ln 2} \)
3. Combine using the product rule
\( h'(x) = \frac{1}{2\sqrt{\ln x}} \cdot \frac{1}{x} \cdot \log_2(x^3 + 1) + \sqrt{\ln x} \cdot \frac{3x^2}{(x^3 + 1) \ln 2} \)
Derivative of exponential functions
Here are the formula for finding the derivative of exponential functions.
Real number \( a \) to the power of \( x \): \( a^x \)
\( \left(a^x\right)’ = a^x \ln a \)
Example:
\( \text{Given function: } 5^{2x + 3} \)
\( \text{Derivative: } \left(5^{2x + 3}\right)’ = 5^{2x + 3} \ln 5 \cdot 2 \)
Real number \( a \) to the power of \( f(x) \): \( a^x \)
\( \left(a^{f(x)}\right)’ = a^{f(x)} \ln a \cdot f'(x) \)
Example:
\( \text{Given function: } 4^{\sin x} \)
\( \text{Derivative: } \left(4^{\sin x}\right)’ = 4^{\sin x} \ln 4 \cdot \cos x \)
\( e \) to the power of \( x \): \( e^x \)
\( \left(e^x\right)’ = e^x \)
Example:
\( \text{Given function: } e^{3x – 2} \)
\( \text{Derivative: } \left(e^{3x – 2}\right)’ = e^{3x – 2} \cdot 3 \)
\( e \) to the power of \( f(x) \): \( e^{f(x)} \)
\( \left(e^{f(x)}\right)’ = e^{f(x)} \cdot f'(x) \)
Example:
\( \text{Given function: } e^{\sqrt{x}} \)
\( \text{Derivative: } \left(e^{\sqrt{x}}\right)’ = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} \)
Here is one more example:
\( \text{Find the derivative of } h(x) = e^{2x} \cdot \ln(x^2) \)
1. Apply the product rule
\( h'(x) = e^{2x} \cdot (\ln(x^2))’ + (e^{2x})’ \cdot \ln(x^2) \)
2. Define the derivatives
\( (\ln(x^2))’ = \frac{2}{x} \)
\( (e^{2x})’ = e^{2x} \cdot 2 \)
3. Combine using the product rule
\( h'(x) = e^{2x} \cdot \frac{2}{x} + e^{2x} \cdot 2 \cdot \ln(x^2) \)
Recommended EffortlessMath Books
For a workbook that nails down log and exponential rules before calculus, the Algebra II for Beginners covers logarithms with worked examples. For full precalc-into-calc prep, the Pre-Calculus for Beginners builds the analytic foundation calculus depends on.
Frequently Asked Questions
What is the formula for the derivative of a logarithmic function?
The derivative of ln x is 1/x for x > 0, and the derivative of logₐ x is 1/(x ln a). When the log contains a function, use the chain rule: (ln u)′ = u′/u and (logₐ u)′ = u′/(u ln a). For example, the derivative of log₂ x is 1/(x ln 2), and since ln 2 ≈ 0.693, that is about 1.443/x.
What are some examples of derivatives of logarithmic functions?
For y = ln(x³ + 2x), the chain rule gives y′ = (3x² + 2)/(x³ + 2x). For y = log₁₀(5x), the 5 cancels: y′ = 5/(5x ln 10) = 1/(x ln 10). For y = ln(sin x), y′ = cos x/sin x = cot x. In each case, find the inner derivative first, then divide by the inside expression.
Are logarithmic functions hard?
Logarithms feel tricky until you see them as the inverse of exponents: log₂ 8 = 3 because 2³ = 8. Their derivatives are actually among the simplest in calculus, since (ln x)′ = 1/x. Most mistakes come from forgetting the chain rule or the domain, because ln x is only defined for x > 0. Practicing the log properties first makes the calculus much easier.
What is a hard logarithmic differentiation example?
A classic is y = xˣ, where the variable is in both the base and the exponent. Take the natural log of both sides: ln y = x ln x. Differentiate: y′/y = ln x + 1. Then multiply by y to get y′ = xˣ(ln x + 1). The same method turns messy products and quotients into sums that are easier to differentiate.
What are the derivatives of logarithmic and exponential functions?
For exponentials, (eˣ)′ = eˣ and (aˣ)′ = aˣ ln a. For logarithms, (ln x)′ = 1/x and (logₐ x)′ = 1/(x ln a). So the derivative of 5ˣ is 5ˣ ln 5, and the derivative of e^(2x) is 2e^(2x) by the chain rule. The ln a factor appears in both families because aˣ = e^(x ln a).
How are derivatives of logarithmic and exponential functions used in real-world problems?
They measure rates of change in growth and decay. If a population follows P(t) = 1000e^(0.05t), then P′(t) = 50e^(0.05t), so at t = 0 it grows by about 50 per year. Log derivatives appear with measurement scales like pH and decibels, and logarithmic differentiation helps find growth rates in finance and science models.
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