Derivatives of Basic Functions

Introduction

This section covers the derivatives of the fundamental functions: \(x^{n}\) for \(n \in Q\), \(e^{x}\), \(\log_{e}(x)\), \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\). These derivatives are essential building blocks for differentiating more complex functions.

Power Rule: \(x^n\) for \(n \in Q\)

The power rule states that if \(f(x) = x^n\), where \(n\) is any rational number, then its derivative is given by:

Examples:

• If \(f(x) = x^3\), then \(f'(x) = 3x^2\).
• If \(f(x) = x^{-2}\), then \(f'(x) = -2x^{-3} = \frac{-2}{x^3}\).
• If \(f(x) = \sqrt{x} = x^{\frac{1}{2}}\), then \(f'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}\).

Exponential Function: \(e^x\)

The derivative of the exponential function \(f(x) = e^x\) is itself:

Natural Logarithm: \(\log_e(x)\) or \(\ln(x)\)

The derivative of the natural logarithm function \(f(x) = \log_e(x) = \ln(x)\) is:

Note: \(x > 0\) for the natural logarithm to be defined.

Trigonometric Functions

Sine Function: \(\sin(x)\)

The derivative of the sine function \(f(x) = \sin(x)\) is the cosine function:

Cosine Function: \(\cos(x)\)

The derivative of the cosine function \(f(x) = \cos(x)\) is the negative sine function:

Tangent Function: \(\tan(x)\)

The derivative of the tangent function \(f(x) = \tan(x)\) is the square of the secant function:

Summary Table

Applications

These derivatives are used extensively in various applications, including:

• Finding the gradient of a curve at a specific point.
• Determining the rate of change of a function.
• Optimization problems (finding maximum and minimum values).
• Modelling physical phenomena.

Examples

1. Find the derivative of \(f(x) = 3x^4 - 2e^x + \ln(x) + 5\sin(x)\).

   \[f'(x) = 12x^3 - 2e^x + \frac{1}{x} + 5\cos(x)\]

2. Find the derivative of \(g(x) = 2\cos(x) - 4\tan(x)\).

   \[g'(x) = -2\sin(x) - 4\sec^2(x)\]