Differentiation

Differentiation

1. Why? To find the gradient of a curve at a point

   • Equivalent to finding the gradient of the tangent to the curve at that point
     • Gradient of equation is change in y divided by change in x
       •           y-y1=m(x-x1)   m=(y-y1) /(x-x1)     
       • Gradient of normal is the negative inverse of m or negative inverse dy/dx
         •   y=x3 at x =1, y=1 dy/dx = 3x^2 so at x=1, gradient = 3.   Normal = - 1/m So at x=1, y=1 gradient = -1/3      
     • Gradient of a tangent= dy/dx
       •      y=x3 at x =1, y=1  dy/dx = 3x^2 so at x=1, gradient = 3.
       • A gradient is the rate of change

2. How to differentiate?

   • Differentiating a polynomial function (one variable)
     • Differenting polynomials
   • Chain Rule
     • Chain Rule Slides
   • Product Rule
     • Product Rule
   • Quotient Rule
     • Quotient Rule
   • Natural Logarithm and Exponential functions
     • Natural Log Function
     • Natural exponential function
   • Trig Functions
     • Trig functions

3. The gradient of a function has different names

   • The gradient function
   • The derived function with respect to x
   • The differential coefficient with respect to x
   • The first differential with respect to x
   • dy/dx
   • f'(x)

4. Differentiate dy/dx to get the second order differential

   • The second order differential has different names
     • d^2y/dx^2
     • f''(x)
     • The second derivative of a function

5. How to find maximum and minimum values of the function

   • At maximum and minimum values of f(x), f'(x) = 0.
     • At maximum value, f''(x) is negative
     • At minimum value, f''(x) is positive

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