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Differentiation — Derivatives of natural logarithmic functions

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Differentiation — Derivatives of Natural Logarithmic Functions

Natural logarithmic functions involve \(\ln x\). Their derivatives are very important in calculus and are often used together with the chain rule, product rule, and quotient rule.

1. The basic derivative

The derivative of \(\ln x\) is:

\[ \frac{d}{dx}(\ln x)=\frac{1}{x} \]

2. Important restriction

The function \(\ln x\) is only defined for \(x>0\).

\[ \ln x \text{ exists only when } x>0 \]

So when differentiating logarithmic functions, always check the domain.

3. Derivative of \(\ln(f(x))\)

If the logarithm contains a function of \(x\), we use the chain rule.

\[ \frac{d}{dx}\bigl(\ln(f(x))\bigr)=\frac{f'(x)}{f(x)} \]

This is one of the most useful logarithmic differentiation rules.

4. Worked example 1

Differentiate

\[ y=\ln x \]

\[ \frac{dy}{dx}=\frac{1}{x} \]

5. Worked example 2

Differentiate

\[ y=\ln(3x+1) \]

Here \(f(x)=3x+1\), so \(f'(x)=3\).

\[ \frac{dy}{dx}=\frac{3}{3x+1} \]

6. Worked example 3

Differentiate

\[ y=\ln(x^2+4) \]

Here \(f(x)=x^2+4\), so \(f'(x)=2x\).

\[ \frac{dy}{dx}=\frac{2x}{x^2+4} \]

7. Worked example 4

Differentiate

\[ y=\ln(\sin x) \]

Here \(f(x)=\sin x\), so \(f'(x)=\cos x\).

Therefore:

\[ \frac{dy}{dx}=\frac{\cos x}{\sin x} \] \[ \frac{dy}{dx}=\cot x \]

8. Logarithms with products, quotients and powers

Before differentiating, it is often useful to simplify using logarithm laws:

\[ \ln(ab)=\ln a+\ln b \] \[ \ln\left(\frac{a}{b}\right)=\ln a-\ln b \] \[ \ln(a^n)=n\ln a \]

This can make differentiation much easier.

9. Worked example 5

Differentiate

\[ y=\ln(x^3) \]

First simplify:

\[ y=3\ln x \]

Then differentiate:

\[ \frac{dy}{dx}=3\cdot\frac{1}{x}=\frac{3}{x} \]

10. Worked example 6

Differentiate

\[ y=\ln\left(\frac{x+1}{x}\right) \]

First use the quotient law:

\[ y=\ln(x+1)-\ln x \]

Differentiate:

\[ \frac{dy}{dx}=\frac{1}{x+1}-\frac{1}{x} \]

11. Using product and quotient rules

Sometimes logarithmic functions appear together with other functions.

Example: Differentiate

\[ y=x\ln x \]

Use the product rule:

\[ \frac{dy}{dx}=x\left(\frac{1}{x}\right)+\ln x(1) \]

\[ \frac{dy}{dx}=1+\ln x \]

12. Common mistakes

  • Forgetting the chain rule in \(\ln(f(x))\).
  • Writing the derivative of \(\ln x\) as \(\ln(1)\), which is incorrect.
  • Ignoring the domain restriction \(x>0\).
  • Not using log laws first when they would simplify the work.

13. Exam tips

  • Remember: \[ \frac{d}{dx}(\ln x)=\frac{1}{x} \]
  • For \(\ln(f(x))\), use \[ \frac{f'(x)}{f(x)}. \]
  • Use log laws to simplify before differentiating if possible.
  • Check the domain carefully.
  • Look out for product rule, quotient rule, and chain rule working together.
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