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

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

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