Derivative of Logarithmic Functions: A Hard Task Made Easy

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:

\( \text{Given function: } \ln (3x) \)

\( \text{Derivative: } \left(\ln (3x)\right)’ = \frac{1}{3x} \cdot 3 = \frac{1}{x} \)

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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

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\( h'(x) = \frac{1}{2\sqrt{\ln x}} \cdot \frac{1}{x} \cdot \log_2(x^3 + 1) + \sqrt{\ln x} \cdot \frac{3x^2}{(x^3 + 1) \ln 2} \)

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

\( h'(x) = e^{2x} \cdot \frac{2}{x} + e^{2x} \cdot 2 \cdot \ln(x^2) \)

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