Differentiation

Differentiation

1. f'(x)=d/dx(xª)=ax^(a-1)

   • Used to find rates of change

2. Finds the gradient at certain points on a graph

   • Can find the equation of a tangent or normal to the curve
     • Tangent - Differentiate and sub in 'x' value to get gradient then work out equation
     • Normal - Differentiate and sub in 'x' value. Find the negative reciprocal to get the gradient then find the equation
   • Stationary Point when gradient equals 0
     • Differentiate, make equal to 0 and solve for x
     • Differentiate twice to find if maximum or minimum
       • Minimum if f''(x) > 0
       • Maximum if f''(x) < 0

3. Differentiate from first principles

   • f′(x) = ((f(x + h) − f(x)) /h) lim h→0

4. Curves can be concave or convex and have points of inflection

   • Convex when f''(x) > 0
   • Concave when f''(x) < 0
   • Point of inflection is when a curve changes from concave to convex

5. Used to find maximum and minimum values for volume and area

   • Differentiate again to prove that it is maximum or minimum

6. Chain rule is used to differentiate functions of functions

7. Differentiating e^x, ln x and a^x