Implicit Differentiation

Unit: Differentiation-Composite, Implicit & Inverse function

Chapter: Implicit Differentiation

Reference: – Implicit Functions, Implicit equations, Derivatives of Implicit relations, Chain rules, Slope of tangent lines, Equation of tangent line, Implicit differentiation, Higher order derivatives, vertical & Horizontal Tangent lines, Curve sketching, Optimization, Trigonometric Functions.

After studying this chapter, you should be able to:

  • Differentiation Rule & Finding Derivatives.
  • High-order Derivatives.
  • Tangent & Normal lines, Related rates.
  • Second derivatives & Concavity

 

Introduction to Implicit Differentiation

  1. Definition of Implicit Differentiation:

 

  • Implicit differentiation is a technique used to find the derivative of an implicitly defined function.
  • An implicitly defined function is given in the form of an equation that may involve multiple variables.
  • Unlike explicit functions, which are solved explicitly for a dependent variable, implicit functions do not isolate the dependent variable.
  1. Differentiation Rules:

 

  • Implicit differentiation involves applying the basic rules of differentiation to both sides of the equation.
  • These rules include the power rule, product rule, quotient rule, and chain rule.
  • When differentiating for a particular variable, all other variables are treated as constants.
  1. Finding Derivatives of Implicit Equations:

 

  • To find the derivative of an implicitly defined function, you differentiate both sides of the equation for the independent variable.
  • When differentiating terms that involve the dependent variable, you apply the chain rule.
  • The derivative of the dependent variable is represented using the notation dy/dx.
  1. Higher-Order Derivatives:

 

  • Implicit differentiation can be extended to find higher-order derivatives of implicitly defined functions.
  • To find the second derivative, you differentiate the first derivative obtained through implicit differentiation.
  1. Tangent Lines and Normal Lines:

 

  • Implicit differentiation allows you to find the equations of tangent lines and normal lines to curves defined implicitly.
  • To find the slope of the tangent line at a given point, you substitute the coordinates of the point into the derivative expression obtained through implicit differentiation.
  • The slope of the tangent line is used to find the equation of the tangent line using point-slope form.
  • The normal line is perpendicular to the tangent line and has a slope that is the negative reciprocal of the tangent line's slope.
  1. Related Rates:

 

  • Implicit differentiation is useful for solving related rate problems.
  • Related rates problems involve finding the rate at which one variable changes for another variable.
  • Implicit differentiation allows you to differentiate both sides of an equation for time (or another variable) and solve for the desired rate.
  1. Second Derivative and Concavity:
  • Implicit differentiation can be used to find the second derivative of implicitly defined functions.
  • The second derivative provides information about the concavity and points of inflection of the curve defined implicitly.
  1. Optimization:

 

  • Implicit differentiation can be applied to optimization problems, where you need to find the maximum or minimum values of a quantity given certain constraints.
  • By setting the derivative of the implicitly defined function equal to zero and solving for the independent variable, you can find critical points that correspond to extreme values.

 

Differentiation Rule & Finding Derivatives:

 

  1. Power Rule:
    • If you have a function of the form f(x) = xn, where n is a constant, the derivative is given by f'(x) = nx(n-1).
    • For example, if f(x) = x2, the derivative is f'(x) = 2x.
  2. Product Rule:
    • If you have two functions, u(x) and v(x), the derivative of their product is given by (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).
    • For example, if f(x) = x2 * sin(x), then f'(x) = (2x * sin(x)) + (x2 * cos(x)).
  3. Quotient Rule:
    • If you have two functions, u(x) and v(x), the derivative of their quotient is given by (u(x)/v(x))' = (u'(x)v(x) – u(x)v'(x))/(v(x))2.
    • For example, if f(x) = (x2 + 1) / x, then f'(x) = ((2x * x) – (x2 + 1))/(x2)2.
  4. Chain Rule:
    • The chain rule is used when you have a composition of functions.
    • If you have a function y = f(g(x)), where g(x) is an inner function and f(u) is an outer function, the derivative is given by dy/dx = f'(g(x)) * g'(x).
    • For example, if f(x) = sin(2x), then f'(x) = 2 * cos(2x).
  5. Trigonometric Functions:
    • The derivatives of common trigonometric functions are:
      • d/dx (sin(x)) = cos(x)
      • d/dx (cos(x)) = -sin(x)
      • d/dx (tan(x)) = sec2(x)
    • Similarly, you can find the derivatives of other trigonometric functions using the chain rule and the quotient rule.
  6. Exponential and Logarithmic Functions:
    • The derivatives of exponential and logarithmic functions are:
      • d/dx (ex) = ex
      • d/dx (ln(x)) = 1x
    • For other exponential and logarithmic functions, you can use the chain rule to find their derivatives.

 

 

Tangent, Normal lines & Related rates: –

Implicit Differentiation & High Order Derivatives: –

 

  1. Implicit Functions: An implicit function is a function where the dependent and independent variables are not explicitly separated. In other words, the equation defining the function does not express y explicitly in terms of x.

 

  1. Implicit Differentiation: Implicit differentiation is a technique used to find the derivatives of implicitly defined functions.

 

  1. Procedure: To perform implicit differentiation, you treat the dependent variable y as a function of x and differentiate both sides of the equation for x. However, when differentiating y for x, you also need to consider the chain rule since y is not explicitly expressed in terms of x.

 

  1. Chain Rule in Implicit Differentiation: When differentiating y for x, you apply the chain rule to the terms involving y. This involves multiplying the derivative of y for x (dy/dx) by the derivative of the term inside the brackets for y.

 

  1. Simplification: After applying implicit differentiation, you can simplify the resulting equation to solve for dy/dx, which represents the derivative of the implicitly defined function.

 

  1. Applications: Implicit differentiation is commonly used in various mathematical fields, such as physics and engineering, where relationships between variables are defined implicitly.

 

Example: – Show that the differential equation (x – y) dy/dx  = x + 2y is homogeneous and solve it.

Solution: (x – y) dy/dx  = x + 2y

                                                … (1)

Let F(x, y) =

Now,

Therefore, F(x, y) is a homogenous function of degree zero. So, the given differential equation is a homogenous differential equation.

Put y = vx                                                              … (2)

Differentiating equation (2) with respect to, x we get

                                                             … (3)

Substituting the value of y and dy/dx  in equation (1) we get

 

Integrating both sides of equation (5), we get

v-1v2+v+1dv=-dxx  

122v+1-3v2+v+1dv=-logx+C1  

122v+1v2+v+1dv-321v2+v+1dv=-logx+C1  

12logv2+v+1–321v+122+322dv=-logx+C1  

12logv2+v+1–32.23tan-12v+13=-logx+C1  

12logv2+v+1+12 logx2=3 tan-12v+13+C1  

Replacing v by y/x  , we get

12logy2x2+yx+1+12 logx2=3 tan-12y+x3x+C1  

12logy2x2+yx+1x2=3 tan-12y+x3x+C1  

log y2+xy+x2=23 tan-12y+x3x+2C1  

log x2+xy+y2=23 tan-1x+2y3x+C  

which is the general solution of the given differential equation.

Key Points

  • The chain rule is fundamental in calculus and used to differentiate composite functions.

 

  • It provides a method to find the derivative of a function that is composed of multiple functions.

 

  • The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

 

  • f'(g(x)) represents the derivative of the outer function f(u) evaluated at the inner function g(x).

 

  • g'(x) represents the derivative of the inner function g(x) for x.

 

  • The chain rule allows us to break down complex functions into simpler functions and differentiate them separately.

 

  • It is essential to apply the chain rule when the function involves compositions of functions like exponentials, logarithms, or trigonometric functions.

 

  • The chain rule can be extended to functions with multiple variables using partial derivatives.

 

  • Implicit differentiation uses the chain rule to find derivatives of implicitly defined functions.

 

  • Understanding and correctly applying the chain rule is crucial for solving differentiation problems and has various applications in mathematics, science, and engineering.

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