**Definitions*** Differentiation* is a way to find the gradient of a function at any point, written as f ′(x), y′ and

^{dy}/

_{dx}.

* Tangent line to a point on a curve* is a linear line with the same gradient as that point on the curve.

**4.1. Polynomials**

**4.2. Tangent and normal**

**Tangent** line with the same gradient as a point on a curve.

**Normal** perpendicular to the tangent m = ^{−1}/_{slope of tangent}

Both are linear lines with general formula: y = mx + c .

1. Use derivative to find gradient of the tangent. For normal then do ^{−1}/_{slope of tangent }.

2. Input the x-value of the point into f(x) to find y.

3. Input y, m and the x-value into y = mx + c to find c.

**4.3. Turning points**

**4.4. Sketching graphs**

Gather information before sketching:

**Intercepts**

x-intercept: f (x) = 0

y-intercept: f (0)

**Turning points**

minima: f ′(x) = 0 and f ′′(x) < 0

maxima: f ′(x) = 0 and f ′′(x) > 0

point of inflection: f ′′(x) = 0

**Asymptotes**

vertical: x-value when the function divides by 0

horizontal: y-value when x → ∞

Plug the found x-values into f (x) to determine the y-values.

**4.5. Applications **

**Kinematics**

Derivative represents the rate of change, integration the reverse.