Differentiation (P1)

This section covers essential differentiation techniques, the analysis of stationary points, tangents and normals to curves, and the concept of rates of change.

1. Differentiation Techniques

1.1 Basic Rules

The power rule is fundamental:

If $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

Other basic rules include:

• Constant Rule: $\frac{d}{dx}(c) = 0$
• Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$
• Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$

1.2 Product Rule

If $y = u(x)v(x)$, then $\frac{dy}{dx} = \frac{du}{dx}v(x) + u(x)\frac{dv}{dx}$.

1.3 Quotient Rule

If $y = \frac{u(x)}{v(x)}$, then $\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{v(x)^2}$.

1.4 Chain Rule

If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

1.5 Implicit Differentiation

Used when $y$ is not explicitly defined as a function of $x$. Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ and solving for $\frac{dy}{dx}$.

2. Stationary Points

2.1 Finding Stationary Points

Stationary points occur where $\frac{dy}{dx} = 0$ or is undefined.

To find stationary points:

1. Find the derivative $\frac{dy}{dx}$.
2. Set $\frac{dy}{dx} = 0$ and solve for $x$. These are critical points.
3. Determine the nature of the stationary points using the second derivative test.

2.2 Second Derivative Test

Let $f(x)$ be a function. If $f'(c) = 0$, then:

Second Derivative	Nature of Stationary Point
$f''(c) > 0$	Local Minimum
$f''(c) < 0$	Local Maximum
$f''(c) = 0$	Test is inconclusive

3. Tangents and Normals

3.1 Gradient of Tangent

The gradient of the tangent to a curve at a point is equal to the value of the derivative at that point.

If the curve is given by $y = f(x)$, then the gradient of the tangent at $x = a$ is $f'(a)$.

3.2 Equation of Tangent

The equation of the tangent to the curve $y = f(x)$ at the point $(a, f(a))$ is given by:

$$y - f(a) = f'(a)(x - a)$$

3.3 Gradient of Normal

The gradient of the normal to a curve at a point is the negative reciprocal of the gradient of the tangent at that point.

If the gradient of the tangent is $m$, then the gradient of the normal is $-\frac{1}{m}$.

3.4 Equation of Normal

The equation of the normal to the curve $y = f(x)$ at the point $(a, f(a))$ is given by:

$$y - f(a) = -\frac{1}{f'(a)}(x - a)$$

4. Rates of Change

4.1 Definition

The rate of change of one quantity with respect to another is the derivative of the dependent variable with respect to the independent variable.

If $y$ is a function of $x$, then the rate of change of $y$ with respect to $x$ is $\frac{dy}{dx}$.

4.2 Applications

Rates of change are used to model real-world situations where quantities are changing with respect to time or another variable. Examples include:

• Velocity and acceleration (displacement vs. time)
• Population growth
• Chemical reactions