Chapter 9 Differentiation

(Q) = small change





1. If y = f (x) , that is a function in terms of x, therefore dy/dx = f' (x).

2. If y = ax^n , therefore dy/dx = nax^(n - 1) , a is a constant.

a) If y = x^n , therefore dy/dx = nx^(-1)

b) If y = k , therefore dy/dx = 0 , k is a constant.

3. The gradient of curve y = f (x) at a certain point is the derivative of y with respect to x , that is dy/dx or f' (x).

4. a) Equation of tangent :

y - b = m1 (x - a) , m1 = f'(a)

b) Equation of normal :

y - b = m2 (x - a) , m2 = -1/m2

5. Differentiation of a product ( Product Rule )

y = uv

dy/dx = (u) dy/dx + (v) du/dx

6. Differentiation of a quotient ( Quotient Rule )

y = u/v

dy/dx = ((v)du/dx - (u)dv/dx) / v^2


7. Differentiation of a composite function (Chain Rule )

y = f (u) and u = g (x) , therefore :

dy/dx = dy/du X du/dx

8. At the turning point or stationary point , dy/dx = 0.

a) Maximum point if d^2y/dx^2 < 0

b) Minimum point if d^2y/dx^2 > 0

9. The related rate of change :

dy/dt = dy/dx X dx/dt

10. Small changes and approximation :

Qy = dy/dx X Qx

11. Second order differentiation :

d^2y/dx^2 = d/dx (dy/dx) = f'' (x)