Further Differentiation

Further Differentiation

1. The Chain Rule

   • The Formula
     • dy/dx = dy/du * du/dx
   • Examples
     • Differentiate y = sin3x
       • y = sin u where u = 3x
         • dy/dx = dy/du *du/dx
           • = cos u * (3)
             • = 3 cos 3x
               • Change u to 3x and simplify
             • dy/du = cos u, du/dx = 3
           • Use the Chain rule
         • This is a composite function
     • Find the equation of the tangent to the graph of y =(3x+5)^2 at the point where x = 2
       • y= u^2, u = 3x+5
         • dy/du = 2u, du/dx = 3
           • dy/dx = dy/du * du/dx
             • = 2u*3 = 6u =6(3x+5) = 18x+30
               • when x =2 18x+30= 66
                 • when x =2 (3x+5)^2 = 121
                   • Tangent at (2,11)
                     • y-y1 = m(x-x1)
                       • y-121=66(x-2)
                         • y=66x -132 +121
                           • y=66x-11
                             • Final answer
                           • simplify
                         • subbing in values
                       • Equation of a line
                   • Finding the y coordinate of the tangent at x = 2 y(2) = (11)^2 = 121
                 • Finding out the gradient
               • Simplifying the expression
             • du cancels to get dy/dx
           • Differentiate each equation
         • Chain Rule
     • A particle Moves in a straight line with displacement at time t from O given by s = 6cos(t/3)m. Find at time t=2π a) the velocity of the particle b) the acceleration of the particle
       • s = 6Cos(u) u = t/3
         • Chain Rule
         • ds/du = -6Sin(u) du/dt = 1/3
           • Differentiate each equation
           • ds/dt = -2sin(t/3)
             • ds/du * du/dt = ds/dt
             • When t = 2π -2Sin(2π/3) = -3^1/2
               • plug in 2π and get an answer
       • -2Sin(u) where u = t/3
         • dv/du = -2Cos(u) du/dt =1/3
           • dv/dt = -2/3Cos(t/3)
             • When t = 2π -2/3Cos(2π/3) = 1/3
               • Plug in 2π and get an answer
             • dv/du * du/dt = dv/dt
           • Differentiate each equation
         • Chain Rule
     • a) solve the equation sin 2x = sin x for 0<x<2π, giving your answers in terms of π
       • 2sinxcosx = sinx
         • 2Cosx
       • b) Find the coordinates of the statio0nairy points of the curve y = sin 2x - sin x for 0<x<2π, giving your answers correct to three signifficant figures
         • c) hence sketch the curve y = sinx - sinx for 0<x<2π