Definition of Differentiation

Differentiation is the process of finding the rate at which a function changes at any point. It is defined as the limit of the average rate of change as the interval approaches zero.

As \(\Delta x \to 0\), the ratio becomes the derivative:

Derivatives of Standard Functions

1. \(\frac{d}{dx}(c) = 0\), where \(c\) is a constant
2. \(\frac{d}{dx}(x) = 1\)
3. \(\frac{d}{dx}(x^n) = nx^{n-1}\)
4. \(\frac{d}{dx}(\sin x) = \cos x\)
5. \(\frac{d}{dx}(\cos x) = -\sin x\)
6. \(\frac{d}{dx}(\tan x) = \sec^2 x\)
7. \(\frac{d}{dx}(e^x) = e^x\)
8. \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)

Derivatives of Composite Functions

1. Addition/Subtraction Rule

If \( y = f(x) \pm g(x) \), then

Example: \(\frac{d}{dx}(\sin x + x^2) = \cos x + 2x\)

2. Product Rule

If \( y = f(x) \cdot g(x) \), then

Example: \(\frac{d}{dx}(x \cdot \sin x) = \sin x + x\cos x\)

3. Quotient Rule

If \( y = \frac{f(x)}{g(x)} \), then

Example: \(\frac{d}{dx}\left(\frac{x}{\sin x}\right) = \frac{\sin x - x\cos x}{\sin^2 x}\)

4. Chain Rule

If \( y = f(g(x)) \), then

Examples:

1. \(\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x\)
2. \(\frac{d}{dx}(\ln(3x+1)) = \frac{1}{3x+1} \cdot 3\)
3. \(\frac{d}{dx}(e^{5x}) = e^{5x} \cdot 5\)
4. \(\frac{d}{dx}((2x+1)^4) = 4(2x+1)^3 \cdot 2\)

Maxima and Minima

In calculus, a function may have turning points where it reaches a maximum or minimum value. These points occur where the derivative is zero, i.e., where:

To determine the nature of the turning point:

1. If \(\frac{d^2y}{dx^2} > 0\), the point is a local minimum (concave up).
2. If \(\frac{d^2y}{dx^2} < 0\), the point is a local maximum (concave down).

The graph below shows an example curve with a local maximum and minimum:

Graph Showing Maximum

A function has a local maximum when it reaches a peak, and the derivative changes from positive to negative. This happens when:

Graph Showing Minimum

A function has a local minimum when it reaches a valley, and the derivative changes from negative to positive. This happens when: