CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
5. CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
You already know about the chain rule whether you have called it that or not. Look at this example.
Example 1. Find dy/dx if y = ln sin 2x. You would say dy
(2x) = 2 cot 2x. dx
(sin 2x) =
We could write this problem as y = ln u,
v = 2x. Then we would say
This is an example of the chain rule. We shall want a similar equation for a function of several variables. Consider another example.
200 Partial Differentiation Chapter 4
Example 2. Find dz/dt if z = 2t 2 sin t.
Differentiating the product, we get
We could have written this problem as z = xy,
where x = 2t 2 and y = sin t, dz
But since x is ∂z/∂y and y is ∂z/∂x, we could also write
We would like to be sure that (5.1) is a correct formula in general, when we are given any function z(x, y) with continuous partial derivatives and x and y are differentiable functions of t. To see this, recall from our discussion of differentials that we had
where ǫ 1 and ǫ 2 → 0 with ∆x and ∆y. Divide this equation by ∆t and let ∆t → 0; since ∆x and ∆y → 0, ǫ 1 and ǫ 2 → 0 also, and we get (5.1). It is often convenient to use differentials rather than derivatives as in (5.1). We would like to be able to use (3.6), but in (3.6) x and y were independent variables and now they are functions of t. However, it is possible to show (Problem 8) that dz as defined in (3.6) is a good approximation to ∆z even though x and y are related. We may then write
whether or not x and y are independent variables, and we may think of getting (5.1) by dividing (5.3) by dt. This is very convenient in doing problems. Thus we could do Example 2 in the following way:
dz = x dy + y dx = x cos t dt + y · 4t dt = (2t 2 cos t + 4t sin t)dt, dz
= 2t 2 cos t + 4t sin t. dt
In doing problems, we may then use either differentials or derivatives. Here is another example.
Section 5 Chain Rule or Differentiating a Function of a Function 201 Example 3. Find dz/dt given z = x y , where y = tan −1 t, x = sin t.
Using differentials, we find dt
2 dt . 1+t You may wonder in a problem like this why we don’t just substitute x and y
= yx y−1 cos t + x y
ln x ·
as functions of t into z = x y to get z as a function of t and then differentiate. Sometimes this may be the best thing to do but not always. For example, the resulting formula may be very complicated and it may save a lot of algebra to use (5.1) or (5.3). This is especially true if we want dz/dt for a numerical value of t. Then there are cases when we cannot substitute; for example, if x as a function of t is given by x+ e x = t, we cannot solve for x as a function of t in terms of elementary functions. But we can find dx/dt, and so we can find dz/dt by (5.1). Finding dx/dt from such an equation is called implicit differentiation; we shall discuss this process in the next section.
Computers can find derivatives, so why should we learn the methods shown here and in the following sections? Perhaps the most important reason is that the techniques are needed in theoretical derivations. However, there is also a practical reason: When a problem involves a number of variables, there may be many ways
to express the answer. (You might like to verify that dz/dt = z(y cot t + ln x cos 2 y) is another form for the answer in Example 3 above). Your computer may not give you the form you want and it may be as easy to do the problem by hand as to convert the computer result. But in problems involving a lot of algebra, a computer can save time, so a good study method is to do problems both by hand and by computer and compare results.
PROBLEMS, SECTION 5
1. Given z = xe −y , x = cosh t, y = cos t, find dz/dt. √
2. Given w = u 2 +v 2 , u = cos[ln tan(p + 1 4 1 π)], v = sin[ln tan(p + 4 π)], find dw/dp. −p 2 −q 3. 2 Given r = e ,p=e s ,q=e −s , find dr/ds.
4. Given x = ln(u 2 −v 2 ), u = t 2 , v = cos t, find dx/dt.
5. If we are given z = z(x, y) and y = y(x), show that the chain rule (5.1) gives
6. Given z = (x + y) 5 , y = sin 10x, find dz/dx.
7. Given c = sin(a − b), b = ae 2a , find dc/da. 8. Prove the statement just after (5.2), that dz given by (3.6) is a good approximation
to ∆z even though dx and dy are not independent. Hint: let x and y be functions independent variables). However, ∆x/∆t is nearly dx/dt for small dt and dt = ∆t
since t is the independent variable. You can then show that
„ dx «
∆x =
+ǫ x dt = dx + ǫ x dt,
dt
202 Partial Differentiation Chapter 4
and a similar formula for ∆y , and get
∂z
∂z
∂y dy + (terms containing ǫ’s) · dt = dz + ǫdt, where ǫ → 0 as ∆t → 0.
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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