Differentiation

Adjunto:

• Differentiation quiz

1. Why? To find the gradient of a curve at a point
   1. Equivalent to finding the gradient of the tangent to the curve at that point
      1. Gradient of equation is change in y divided by change in x

         Nota:

         •           y-y1=m(x-x1)   m=(y-y1) /(x-x1)     
         1. Gradient of normal is the negative inverse of m or negative inverse dy/dx

            Nota:

            •   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      
      2. Gradient of a tangent= dy/dx

         Nota:

         •      y=x3 at x =1, y=1  dy/dx = 3x^2 so at x=1, gradient = 3.
         1. A gradient is the rate of change
2. How to differentiate?
   1. Differentiating a polynomial function (one variable)

      Adjunto:

      • Differenting polynomials
   2. Chain Rule

      Adjunto:

      • Chain Rule Slides
   3. Product Rule

      Adjunto:

      • Product Rule
   4. Quotient Rule

      Adjunto:

      • Quotient Rule
   5. Natural Logarithm and Exponential functions

      Adjunto:

      • Natural Log Function
      • Natural exponential function
   6. Trig Functions

      Adjunto:

      • Trig functions
3. The gradient of a function has different names
   1. The gradient function
   2. The derived function with respect to x
   3. The differential coefficient with respect to x
   4. The first differential with respect to x
   5. dy/dx
   6. f'(x)
4. Differentiate dy/dx to get the second order differential
   1. The second order differential has different names
      1. d^2y/dx^2
      2. f''(x)
      3. The second derivative of a function
5. How to find maximum and minimum values of the function
   1. At maximum and minimum values of f(x), f'(x) = 0.
      1. At maximum value, f''(x) is negative
      2. At minimum value, f''(x) is positive