Differentiation

Differentiation

1. dy/dx

   • also known as the gradient function.
     • e.g after differentiating, 4x+1 will be the gradient function.
       • therefore, y will be the numerator and x the denominator.
   • Constant Multiple Rule: d/dx[kf(x)] = k•f(x)

2. Power Rule

   • dy/dx 4x^2 = (2)4x^1 [seen as, power bring down power minus one]

3. Chain Rule

   • equation: dy/dx = dy/du • du/dx

4. Sum and Difference Rule

   • equation: d/dx (u+v) = u•dv/dx + v•du/dx

5. Tangent and Normal

   • equation of tangent line is (Y - Y1) = dy/dx (X - X1)
     • Y1, X1, dy/dx can be found through the question, or by subbing either coordinates.
       • remember to expand the equation so it can look something like: y = 3x + 18
   • equation of normal is (Y - Y1) = -1/tangent gradient (X - X1)

6. Connected rates of change

   • dy/dt = dy/dx • dx/dt
     • question phrased is: "find the rate of change of y, given that the rate of change of x is 2 units/s.
       • rate of change of y = dy/dt if rate of change of x is 2, dx/dt = 2

7. First Derivative Test & Second Derivative Test

   • FDT table [x, sign of dy/dx, sketch of tangent]
     • First Derivative Test
       • is to find maximum, minimum points or point of inflexion.
         • if question says stationary points, dy/dx = 0
           • minimum point: dy/dx > 0 maximum point: dy/dx < 0
   • Second Derivative Test
     • equation is d^2y / dx^2

8. Trigonometric Functions

   • dy/dx of: (sin x ) = cos x (cos x ) = -sin x (tan x) = sec^2 x
     • remember, for logarithmic, exponential, trigonometric functions: remember to use chain rule if x is e.g. 2x+1

9. Exponential Functions

   • dy/dx of: e^x = e^x

10. Logarithmic Functions

    • dy/dx of: ln x = 1/x