Chapter 1. calc_tutorial_14_6_014

1.1 Problem Statement

{3,5,7}
{3,4,5,6}
{2,4}
4*$b-8
4*$c
4*pow($a,3)+$d*$a+$e

Use the Chain Rule to evaluate the partial derivative at the point specified.

(∂g / ∂s) at s = $a, where g(x, y) = x2 − y2, x = s2 + $b, y = $c − 2s.

1.2 Step 1

Question Sequence

Question 1.1

Recall the General Version of the Chain Rule for functions of n variables. Let f(x1, , xn) be a differentiable function of n variables. Suppose that each of the variables x1, ..., xn is a differentiable function of m independent variables t1, ..., tm. Then we have the following for k = 1, ..., m.

(∂f / ∂tk) = (∂f / ∂x1) · (∂x1 / ∂tk) + (∂f / ∂x2) · (∂x2 / ∂tk) + ... + (∂f / ∂xn) · (∂xn / ∂tk)

We wish to find (∂g / ∂s) at s = $a where g(x, y) is a differentiable function of two variables, x and y, and both x and y are differentiable functions of one variable, s.

Therefore we will apply the chain rule in the following way to calculate (∂g / ∂s).

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

Note we have used the notation (dx / ds) and (dy / ds) instead of (∂x / ∂s) and (∂y / ∂s) since x and y are functions of only one variable, s. Otherwise, we would keep the notation for partial derivatives.

Find (∂g / ∂x) and (∂g / ∂y) in terms of x and y given that g(x, y) = x2 − y2.

(∂g / ∂x) =

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

(∂g / ∂y) =

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
Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Find (dx / ds) and (dy / ds) given that x = s2 + $b and y = $c − 2s.

(dx / ds) =

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

(dy / ds) =

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
Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Find (∂g / ∂s) in terms of only s by using the information above.

(∂g / ∂s) = (∂g / ∂x) · (dx / ds) + (∂g / ∂y) · (dy / ds)

= (2x)(2s) + (−2y)(−2)

= (2(s2 + $b))(2s) + (-2($c − 2s))(-2)

= h4XZagboIgc=s3 + U2GIbglD1oM=s + 5dWJhLEUU4Y=

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Find (∂g / ∂s) at s = $a by substituting s = $a into the expression for (∂g / ∂s).

or6dmYWrEbA=

Correct.
Incorrect.