Maths Differentiation

• Differentiate the function
• Find the gradient of the curve to that point (using dy/dx)
• Use this to deduce the gradient, m, of the tangent or normal:
  • gradient of the tangent = gradient of the curve
  • gradient of the normal = 1/gradient of the curve
• Write out the equation y - y1 =m(x-x1), using the points given.
• Simplify to find the equation (or don't if a form is not given - you get full marks).

• Stationary points occur when the gradient is zero. 
• To find the stationary points of a graph;
  • Differentiate the expression f(x) (using dy/dx)
  • Set f'(x) = 0 
  • Solve f'(x) = 0 to find the x values
  • Put the x values back into the original equation to find the y values

• Once you know where the stationary points are, you may have to determine whether they are maximum or minimum points.
  • If d^2y/dx^2 < 0, its a maximum. 
  • If d^2y/dx^2 > 0, its a minimum.

• You can find out if a function is increasing or decreasing at a given point
  • A function is increasing if the gradient is positive - dy/dx > 0 
    • y gets bigger as x gets bigger
  • A function is decreasing if the gradient is negative - dy/dx < 0 
    • y gets smaller as x gets bigger

• You can tell how quickly a function is increasing or decreasing by looking at the size of the gradient;
  • A small increase in x and a large increase in y means a large positive gradient
  • A large increase in x and a small increase in y means a small positive gradient
  • A large increase in x and a small decrease in y means a small negative gradient
  • A small increase in x and a large decrease in y means a large negative gradient

• Quadratic graphs
  • Need the shape, and the points of intersection
    • If the coefficient of x^2 is positive - the graph will be u shaped
    • If the coefficient of x^2 is negative - the graph will be n shaped
    • To find the intercepts of the graph;
      • To find the y- intercept - let x=0 and calculate the value of y (substitute x into equation)
      • To find the x- intercept - let y=0 and solve the equation 0=ax^2+bx+c to find the value or values of x.
  • Vertex points 
    • Complete the square and interpret
      • A function of the form y=p(x+q)^2 + r has a vertec at (-q, r).
        • If p>0, the graph is u-shaped, so the vertex is a minimum.
        • If p<0, the graph is n-shaped, so the vertex is a maximum.
    • Quadratic graphs are always symmetrical, with a vertical line of symmetry that goes through the vertex. You can work out the equation of a line of symmetry by looking at the coordinates of the vertex points - for a quadratic with a vertex of (m,n), the line of symmetry is at x=m. When completing the square, the line of symmetry is at x=-q.

• If the function has no real roots, the shape and the axis intercepts won't be enough to draw the graph. In these cases you'll have to find the coordinates of the vertex point, even if the question doesn't ask you to.
  • Use coefficient to find shape, make x=0 to find the y-intercept, make y=0 and solve to find the x axis intercepts (b^2 - 4ac), and then find minimum/maximum to find the vertex points.
• If the root has two distinct roots - use symmetry of quadratic graphs 
  • The graph of a quadratic function is symmetrical, so the x-coordinate of the vertex is halfway between the roots of the function.
  • Work out the x-value halfway between the two roots and put it into the function to find the corresponding y-value of the vertex. 
• If the function has two equal roots (i.e. one root) - the vertex is at the root
  • If a function has one root, then its graph just touches the x-axis at the root - this point will always be the vertex. 

• Cubic functions has an x^3 term in them - as the highest power of x.
• They can be written as y=ax^3 + bx^2 + cx + d (for when a is not 0)
• All the cubic graphs have a characteristic 'wiggle' 

• y = x^3 - a positive coefficient of x^3 gives a 'bottom-left to top-right' shape.
• y = -x^3 - a negative coefficient of x^3 gives a 'top-left to bottom-right' shape.

Both of these graphs 'flattern out' at the point when the curve starts bending the other way. 

• y= x^3 = x^2 - if the cubic has x^2 terms there'll be more of a 'wiggle'.
• y=5+x-x^2-x^3  - you can tell that this graph has a negative x^3 term because it's got the 'top-left to bottom-right' shape.

• May need to sketch cubics - find where the graph crosses the axes (particularly the x-axis).