Differentiation notes

This section contains useful notes for differentiation.

• The gradient of a curve y=f(x) at a specific point is equal to the gradient of the tangent to the curve at that point.
• The gradient of the tangent at any particular point is the rate of change of y with respect to x.
• The gradient formula for y=f(x) is given by the equation gradient = f’(x) where f’(x) is called the derived function.
• If f(x) = xⁿ, then f’(x) = nx^n-1
• The gradient of a curve can also be represented by dy/dx.
• dy/dx is called the derivative of y with respect to x and the process of finding dy/dx when y is given is called differentiation.
• y = f(x), dy/dx = f’(x)
  
• y=xⁿ, dy/dx = nx^n-1 for all real values of n.
• It can also be shown that if y = axⁿ where a is a constant, then;
  
• If y=f(x) ± g(x) then…
  
• A second order derivative is written as;
  
• You can find the rate of change of a function f at a particular point by using f’(x) and substituting in the value of x.
• The equation of the tangent to the curve y=f(x) at point A, (a, f(a))
  
• The equation of the normal to the curve y=f(x) at point A, (a, f(a)) is;