DERIVATIVES OF THE INVERSE TRIGONOMETRIC FUNCTIONS - Differentiation of Transcendental Functions - The Calculus Primer

The Calculus Primer (2011)

Part V. Differentiation of Transcendental Functions

Chapter 19. DERIVATIVES OF THE INVERSE TRIGONOMETRIC FUNCTIONS

5—14.The Derivative of Arc Sin v and Arc Cos v. It will be recalled that the inverse trigonometric functions may be written as follows:

ifα = sin β,thenβ = arc sin α;

ifα = cos β, thenβ = arc cos α; etc.

Now let

y = arc sin v,

orv = sin y.(1)

Differentiating equation (1) with respect to y by [6] (§5—9):

images

From equation (2), by using the formula for the derivative of an inverse function (§5—2), we have:

images

But v is a function of x; hence, using the formula for the derivative of a function of a function (§5—1):

images

Substituting in (4) from (3):

images

From trigonometry

images

Substituting in (5):

images

By proceeding in the same manner, the reader may verify the formula for the derivative of arc cos v:

images

5—15.Derivative of Arc Tan v and Arc Cot v. The general procedure is the same as for the derivation given in §5—14.

Lety = arc tan v,

orv = tan y.

Differentiating with respect to y:

images

By trigonometry, sec2 y = 1 + tan2 y = 1 + v2. Therefore

images

In the same manner, the reader may verify

images

5—16.Derivative of Arc Sec v and Arc Csc v.

Lety = arc sec v,

orv = sec y.

Differentiating with respect to y:

images

By trigonometry,

images

and sec y = v, by hypothesis.

images

In the same manner, the reader may verify

images

EXAMPLE 1.Differentiate y = arc sin 2x.

images

EXAMPLE 2.Differentiate y = arc tan images.

images

EXERCISE 5—6

Differentiate:

1. y = arc sin x2

images

4. y = x arc sin x

images

5—17.Summary of Formulas. For the reader’s convenience, we summarize below the formulas for differentiating transcendental functions, as was done for algebraic functions in §3—8.

images

images

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5—18.Successive Differentiation. When discussing velocity and acceleration in §4—7, we learned how to find successive derivatives of algebraic functions. We shall now apply this idea to transcendental functions.

EXAMPLE 1.Find the fourth derivative of y = ex + 2x3.

images

EXAMPLE 2.Find the third derivative of y = sin 2x.

images

5—19.Successive Differentiation of Implicit Functions. Suppose we wish to find the second derivative of y with respect to x in the equation of the ellipse b2x2 + a2y2 = a2b2. We proceed to differentiate with respect to x:

images

Now differentiate again, remembering that y is a function of x:

images

Substituting the value of images from (1) in equation (2):

images

But a2y2 + b2x2 = a2b2; therefore images

EXAMPLE 1.Find the second derivative with respect to x of

x2 + y = xy.

images

Differentiating again:

images

Substituting for images from (1):

images

EXAMPLE 2.Find the second derivative with respect to x of

ex = sin y.

images

Differentiating again:

images

Substituting the value of images from (1):

images

5—20.The nth Derivative of a Product. It is sometimes useful to express the nth derivative of the product of two variables in terms of the variables and their successive derivatives. For example, if u and v are functions of x,then

images

By differentiating again, the reader can verify that

images

By mathematical induction, it can be shown, in general, that

images

This is known as Leibniz’s Formula for the nth derivative of a product. It will be seen that the numerical coefficients follow the same law as those of the binomial theorem, and that the indices of the derivatives correspond to the exponents in the binomial expansion. The correspondence can be made complete as follows: just as the first derivatives images and images can be considered as images and images, so u and v (the variables themselves) may be considered as imagesand images.

EXERCISE 5—7

1. Find the second derivative of

y = ex sin x.

2. Find the third derivative of

y = log sin x.

3. Find the fourth derivative of

(a) y = x3 log x

(b) y = sin ax

4. Find images for the equation x2 + y2 = k2.

5. Find images for y2 = 4px.

6. Find images for y2 + x = y.

7. Find, by Leibniz’s formula, the third derivative of y = exx2. (Let u = ex, v = x2.)

EXERCISE 5—8

Review

Differntiate:

images

2. y = et(1 + t2)

3. y = log (2 – 3x2)

4. y = cez

5. y = log (log x)

6. y = elog x2

images

8. y = ex log x

9. y = ecos 2x

10. y = log sin x

11. y = tan x + tan2 x

images

13. y = ex sin x

images

16. y = log cos2 x

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