IMPLICIT DIFFERENTIATION
6. IMPLICIT DIFFERENTIATION
Some examples will show the use of implicit differentiation.
Example 1. Given x + e x = t, find dx/dt and d 2 x/dt 2 .
If we give values to x, find the corresponding t values, and plot x against t, we have a graph whose slope is dx/dt. In other words, x is a function of t even though we cannot solve the equation for x in terms of elementary functions of t. To find dx/dt, we realize that x is a function of t and just differentiate each term of the equation with respect to t (this is called implicit differentiation). We get
Solving for dx/dt, we get
Alternatively, we could use differentials here, and write first dx+e x dx = dt; dividing by dt then gives (6.1).
We can also find higher derivatives by implicit differentiation (but do not use differentials for this since we have not given any meaning to the derivative or dif- ferential of a differential). Let us differentiate each term of (6.1) with respect to t; we get
Solving for d 2 x/dt 2 and substituting the value already found for dx/dt, we get
This problem is even easier if we want only the numerical values of the derivatives at a point. For x = 0 and t = 1, (6.1) gives
and (6.2) gives
2 dt 2 =− . 8 Implicit differentiation is the best method to use in finding slopes of curves with
complicated equations.
Section 7 More Chain Rule 203 Example 2. Find the equation of the tangent line to the curve x 3 − 3y 3 + xy + 21 = 0 at
the point (1, 2).
We differentiate the given equation implicitly with respect to x to get
Substitute x = 1, y = 2:
35 7 Then the equation of the tangent line is
By computer plotting the curve and the tangent line on the same axes, you can check to be sure that the line appears tangent to the curve.
PROBLEMS, SECTION 6
1. If pv a = C (where a and C are constants), find dv/dp and d 2 v/dp 2 .
2. If ye xy = sin x find dy/dx and d 2 y/dx 2 at (0, 0).
3. If x y =y x , find dy/dx at (2, 4).
4. If xe y = ye x , find dy/dx and d 2 y/dx 2
5. If xy 3 − yx 3 = 6 is the equation of a curve, find the slope and the equation of the tangent line at the point (1, 2). Computer plot the curve and the tangent line on the same axes.
6. In Problem 5 find d 2 y/dx 2 at (1, 2).
7. If y 3 −x 2 y = 8 is the equation of a curve, find the slope and the equation of the tangent line at the point (3, −1). Computer plot the curve and the tangent line on the same axes.
8. In Problem 7 find d 2 y/dx 2 at (3, −1).
√ 9. For the curve x 2/3 +y 2/3
√ = 4, find the equations of the tangent lines at (2
2, −2 2), at (8, 0), and at (0, 8). Computer plot the curve and the tangent lines on the same axes.
10. For the curve xe y + ye x = 0, find the equation of the tangent line at the origin. Caution: Substitute x = y = 0 as soon as you have differentiated. Computer plot the curve and the tangent line on the same axes.
11. In Problem 10, find y ′′ at the origin.
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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