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 function 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
y f (x) 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