Derivative of Logarithmic and Exponential Functions

Sahabat Smaridasa - Mathematical Concepts

Previously, we studied "Derivatives of Algebraic Functions" and "Derivatives of Trigonometric Functions". In this article, we will discuss Derivatives of Logarithmic and Exponential Functions. To determine these derivatives, we also need material on "Special Infinite Limits", the "Chain Rule", and the definition and properties of logarithms for proofs.

Derivative of Logarithmic Functions

The simplest logarithmic function is \(y = a \log x\) (with base \(a\) and argument \(x\)). The derivatives are:

• i) \(y = {}^a \log x \quad \Rightarrow \quad y' = \frac{1}{x} \cdot {}^a \log e\)
• ii) \(y = {}^a \log g(x) \quad \Rightarrow \quad y' = \frac{g'(x)}{g(x)} \cdot {}^a \log e\)

where \(e \approx 2.7182818\ldots\) (Euler's number).

📘 Example 1 : Derivatives of Logarithmic Functions

Find the derivatives of the following logarithmic functions:

a) \(y = {}^2 \log x\)    b) \(y = {}^2 \log(2x^3 - x^2 + x - 7)\)    c) \(y = {}^{(2x+1)} \log(x - 2)\)

Solution:

a) Using rule i: \(\displaystyle y' = \frac{1}{x} \cdot {}^2 \log e\).

b) Let \(g(x) = 2x^3 - x^2 + x - 7\) ⇒ \(g'(x) = 6x^2 - 2x + 1\).

\[ y' = \frac{6x^2 - 2x + 1}{2x^3 - x^2 + x - 7} \cdot {}^2 \log e. \]

c) Using the logarithm property \( \log_b a = \frac{\log a}{\log b} \) (with base 10, though any base works):

\( y = {}^{(2x+1)} \log(x - 2) = \frac{\log(x-2)}{\log(2x+1)} \).

Let \(U = \log(x-2)\), \(V = \log(2x+1)\). Then \(U' = \frac{1}{x-2} \log e\), \(V' = \frac{2}{2x+1} \log e\).

\[ y' = \frac{U'V - UV'}{V^2} = \frac{\frac{1}{x-2} \log e \cdot \log(2x+1) - \log(x-2) \cdot \frac{2}{2x+1} \log e}{(\log(2x+1))^2}. \]

Derivative of the Natural Logarithm (ln)

The natural logarithm \(\ln x\) is a logarithm with base \(e\): \(\ln x = e \log x\). Properties of \(\ln\) are the same as logarithm properties.

• i) \(y = \ln x \quad \Rightarrow \quad y' = \frac{1}{x}\)
• ii) \(y = \ln g(x) \quad \Rightarrow \quad y' = \frac{g'(x)}{g(x)}\)

Proof: From the general logarithmic derivative \(y = a \log x \Rightarrow y' = \frac{1}{x} \cdot a \log e\). Since \(a \log e = 1\) when \(a = e\), we get \(y' = \frac{1}{x}\).

📘 Example 2 : Derivatives of Natural Logarithm (ln)

Find the derivatives:

a) \(y = \ln x\)    b) \(y = \ln(x^2 - 3x + 1)\)

Solution:

a) \(\displaystyle y' = \frac{1}{x}\).

b) Let \(g(x) = x^2 - 3x + 1\) ⇒ \(g'(x) = 2x - 3\). Then \(\displaystyle y' = \frac{2x - 3}{x^2 - 3x + 1}\).

Derivative of Exponential Functions

Exponential function derivatives:

• i) \(y = a^x \quad \Rightarrow \quad y' = a^x \ln a\) (where \(\ln a = e \log a\))
• Special case: \(y = e^x \quad \Rightarrow \quad y' = e^x \ln e = e^x \cdot 1 = e^x\)
• ii) \(y = a^{g(x)} \quad \Rightarrow \quad y' = g'(x) \cdot a^{g(x)} \cdot \ln a\)
• Special case: \(y = e^{g(x)} \quad \Rightarrow \quad y' = g'(x) \cdot e^{g(x)}\)

Note: \(\ln e = 1\) by logarithm properties, and \(e \approx 2.7182818\ldots\).

📘 Example 3 : Derivatives of Exponential Functions

Find the derivatives:

a) \(y = 2^x\)    b) \(y = e^x\)    c) \(y = 3^{3x^2 - 2x + 1}\)    d) \(y = e^{3x^2 - 2x + 1}\)

Solution:

a) \(y' = 2^x \ln 2\).

b) \(y' = e^x\).

c) Let \(g(x) = 3x^2 - 2x + 1\) ⇒ \(g'(x) = 6x - 2\). Then \(y' = (6x - 2) \cdot 3^{3x^2 - 2x + 1} \cdot \ln 3\).

d) Let \(g(x) = 3x^2 - 2x + 1\) ⇒ \(g'(x) = 6x - 2\). Then \(y' = (6x - 2) \cdot e^{3x^2 - 2x + 1}\).

Note: These derivative rules are fundamental for differentiating logarithmic and exponential functions, which appear frequently in calculus, particularly in growth and decay models, compound interest, and many other applications.