Learning Development Service Differentiation 2 To book an appointment email: [email protected] Who is this workshop for? • Scientists-physicists, chemists, social scientists etc. • Engineers • Economists • Calculus is used almost everywhere! Learning Development Service What we covered last time : • What is differentiation? Differentiation from first principles. • Differentiating simple functions • Product and quotient rules • Chain rule • Using differentiation tables • Finding the maxima and minima of functions Learning Development Service What we will cover this time: • Second Order Differentiation • Maxima and Minima (Revisited) • Parametric Differentiation • Implicit Differentiation Learning Development Service The Chain Rule by Example (i) Differentiate : Set: The differential is then: Learning Development Service The Chain Rule: Examples Differentiate: 2 1/3 (i) y x 3x 1 (ii) y sincos x (iii) y cos3x 1 3 2x 1 (iv) y 2x 1 Learning Development Service Revisit: maxima and minima Learning Development Service Finding the maxima and minima Learning Development Service The second derivative • The second derivative is found by taking the derivative of a function twice: dy d 2 y y f (x) f (x) f (x) dx dx2 • The second derivative can be used to classify a stationary point Learning Development Service The second derivative • If x0 is the position of a stationary point, the point is a maximum if the second derivative is <0 : d 2 y 0 dx2 and a minimum if: d 2 y 0 dx2 • The nature of the stationary point is inconclusive if the second derivative=0. Learning Development Service The second derivative: Examples 3 • Find the stationary points of the function f (x) x 6x and classify the stationary points. Learning Development Service The second derivative: Examples 1 • Find the local maxima/minima of the function f (x) x4 x3 3 Learning Development Service Parametric Differentiation • Sometimes the equation of a curve is not given in Cartesian form y f ( x ) and instead is given in parametric form: x f (t) y f (t) • Thus, the x and y coordinates are given in terms of another variable, t • To differentiate find the derivative of x with respect to t and y with respect to t • The derivative f ( x ) is then found by dividing the two derivatives Learning Development Service Parametric Differentiation • Thus the derivative is: dy dy dx f (x) dx dt dt 2 • Example 1: find f (x) if x 3t, y t 4t 1 • Example 2: find f (x) if x cost, y sint • Example 3: find if x 3t 4sint, y t 2 t cost Learning Development Service Implicit Differentiation • Equations such as y x2 and y 1/ x are said to define y as a yfunction f (x) of x explicitly i.e. variable y alone on one side of the equation • For example, the equation yx y 1 x is not in the form but can easily be rearranged to take this form • We say the above equation is defined implicitly as a function of x when in the form and is defined explicitly when rearranged and rewritten in the form: x 1 y x 1 Learning Development Service Implicit Differentiation • Implicit differentiation allows an expression to be differentiated even when y cannot be expressed explicitly in terms of x • Example 1: find f (x ) if xy 1 2 3 • Example 2: find if x y 1 y • Example 3: find if 2y x2 sin y Learning Development Service .