April 14, 2024

Table of contents & cheatsheet (DIFFERENTIATION)

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

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

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.

Kinematics