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Mathematics LibreTexts

4.4: Applications of the Power Chain Rule.

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The Power Chain Rule

PCR:[un(t)]=n un1(t)×u(t)=n un1(t) u(t)

greatly expands the diversity and interest of problems that we can analyze. Note that the second form omits the × symbol; multiplication is implied by the juxtaposition of symbols. The general chain rule also will be written

[G(u(x))]=G(u(x)) u(x)

Some introductory examples follow.

Example 4.4.1

  1. Find the slope of the tangent to the circle, x2+y2=13, at the point (2,3). See Figure 4.4.1.
  2. Also find the slope of the tangent to the circle, x2+y2=13, at the point (3,-2).

4-3.JPG

Figure 4.4.1: Graph of the circle x2+y2=13 and tangents drawn at the points (2,3) and (3,-2) of the circle.

Solution First check that 22+32=4+9=13, so that (2,3) is indeed a point of the circle. Then solve for y in x2+y2=13 to get

y1=13x2

Then the Power Chain Rule with n=12 yields

y1=[13x2]=[(13x2)12] (i) Symbolic identity =12(13x2)121×[13x2] (ii) PCR, n=1/2 u=1x2=12(13x2)12×([13][x2]) (iii) Sum Rule =12(13x2)12×(02x) (iv) Constant and Power Rules =x(13x2)12

Observe the exaggerated ( )'s in steps (iii) and (iv). Students tend to omit writing them, but they are necessary. Without the ( )’s, steps (iii) and (iv) would lead to

y1=12(13x2)12×[13][x2](iii) Sum Rule=12(13x2)12×02x(iv) Constant and Power Rules=2x

a notably simpler answer, but unfortunately incorrect. Always:

Second Notice.

Use parentheses, ( )'s, they are cheap.

To finish the computation, we compute y1 at x=2 and get

\(y ^{\prime} _{1} (2) = (−2) (13 − 2 ^{2})^{- \frac{1}{2}} = - \frac{2}{3}\]

and the slope of the tangent to x2+y2=13 at (2,3) is 2/3. An equation of the tangent is

y3x2=23ory=23x+413

B. Now we find the tangent to the circle x2+y2=13 at the point (3,-2). Observe that 32+(2)2=4+9=13 so that (2,-3) is a point of x2+y2=13, but (3,-2) does not satisfy

y1=(13x2)12

because

2(13(3)22)12=2

For (3,-2) we must use the lower semicircle and

y2=(13x2)12

Then

y2=[(13x2)12]=12(13x2)121[1x2]=12(13x2)12(2x)

At x=3

y2(3)=12(1332)12(2×3)=32

and the slope of the line drawn is 3/2.

It is often helpful to put the denominator of a fraction into the numerator with a negative exponent. For example:

Problem. Compute P(t) for P(t)=5(1+t)2. Solution 

P(t)=[5(1+t)2]=[5(1+t)2]=5[(1+t)2]=5((2)(1+t)3)×[(1+t)]=5(2)(1+t)3×1=10(1+t)3

aaa We will find additional important uses of the power rules in the next section.

 

Exercises for Section 4.4, Applications of the Power Chain Rule.

Exercise 4.4.1 Compute y(x) for

  1. y=2x35
  2. y=2x2
  3. y=5(x+1)2
  4. y=(1+x2)0.5
  5. y=1x2
  6. y=(1x2)0.5
  7. y=(2x)4
  8. y=(3x2)4
  9. y=17x2
  10. y=(1+(x2)2)2
  11. y=(1+3x)1.5
  12. y=116x2
  13. y=9(x4)2
  14. y=(9x2)1.5
  15. y=313x

Exercise 4.4.2 Shown in Figure Ex. 4.4.2 is the ellipse,

x218+y28=1

and a tangent to the ellipse at (3,2).

  1. Find the slope of the tangent.
  2. Find an equation of the tangent.
  3. Find the x- and y-intercepts of the tangent.

4-4-2.JPG

Figure for Exercise 4.4.2 Graph of the ellipse x2/18+y2/8=1 and a tangent to the ellipse at the point (3,2).

Exercise 4.4.3 Shown Figure Ex. 4.4.3 is the ellipse,

2x235+3y235=1

and tangents to the ellipse at (2, 3) and at (4, -1).

  1. Find the slopes of the tangents.
  2. Find equations of the tangents.
  3. Find the point of intersection of the tangents.

4-4-3.JPG

Figure for Exercise 4.4.3 Graph of the ellipse 2x2/35+3y2/35=1 and tangents to the ellipse at the points (2,3) and (4,-1).

Exercise 4.4.4 Shown in Figure Ex. 4.4.4 is the circle with center at (2,3) and radius 5. find the slope of the tangent to this circle at the point (5,7). An equation of the circle is

(x2)2+(y3)2=52

4-4-4.JPG

Figure for Exercise 4.4.4 Graph of the circle (x2)2+(y3)2=52 and tangent to the circle at the point (5,7).


4.4: Applications of the Power Chain Rule. is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by LibreTexts.

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