﻿ ﻿DERIVATIVES OF EXPONENTIAL FUNCTIONS - Differentiation of Transcendental Functions - The Calculus Primer

## The Calculus Primer (2011)

### Chapter 17. DERIVATIVES OF EXPONENTIAL FUNCTIONS

5—6. Differentiating the Simple Exponential Function. Let us consider first the simple exponential function y = av, where a is a constant greater than zero (that is, positive), and v is a function of x. Taking the logarithm of both sides to the base e, we have:

log y = v log a, Differentiating with respect to y: Applying the formula for the derivative of an inverse function, §5—3, equation : Now we apply the formula for the derivative of a function of a function, §5—1, equation , since v is a function of x; that is Substituting the value of from (1) above, we get: As a special case, when a = e, log a = log e = 1; hence RULE. The derivative of a constant with a variable exponent is equal to the product of the natural logarithm of the constant, the constant with the variable exponent, and the derivative of the exponent.

EXAMPLE 1.Find (a4x+1).

Solution. (a4x+1) = log a(a4x+1)(4),or4 log a(a4x+1).

EXAMPLE 2.Differentiate y = e−2x.

Solution. = e−2x(−2) = −2e−2x.

EXAMPLE 3.Differentiate s = et3+1.

Solution. EXAMPLE 4.Find in the equation ρ = aθ2.

Solution. = log a·aθ2·2θ.

EXAMPLE 5.Differentiate y = elog x.

Solution. But elog x = x, by the definition of a logarithm.

Therefore EXAMPLE 6.Differentiate y = xe2/x.

Solution. EXERCISE 5—2

Differentiate:

1. y = k4x 3. y = ekx

4. y = kex

5. y = e5−3t  11. ρ = aθ

12. ρ = alogθ

13. ρ = e

14. y = e1/x

15. y = xex

16. y = x2e2x

17. y = xn + nx

18. y = axxa

19. y = ex(x2 − 2x + 2)

20. y = ex(x − 1)

5—7. Differentiating the General Exponential Function. We now consider the more general exponential function y = uv, that is, a function raised to a variable power instead of a constant raised to a variable power. The only restriction is that u shall assume only positive values.

Lety = uv.(1)

Take the logarithm of both sides to the base e:

loge y = v loge u,

or, by the definition of a logarithm,

y = ev log u.(2)

Differentiating equation (2) by formula [4a]: But ev log u = y from equation (2); and y = uv from equation (1); hence ev log u = uv.

Also, by differentiating as a product, Making these substitutions in equation (3), we get: RULE. The derivative of a function with a variable exponent is equal to the sum of the two results obtained by first regarding the exponent as a constant and differentiating, and then by regarding the function as a constant and differentiating.

EXAMPLE 1.Differentiate y = xx.

Solution. EXAMPLE 2.Differentiate Solution.Here u = x, v = ex. EXAMPLE 3.Prove that the formula holds for all values of the constant n by setting v = n in formula .

Solution.If v = n, we have, from : EXERCISE 5—3

Differentiate:

1. y = xx+1

2. y = (x2 + 1)x

3. y = (x3)x

4. y = x2x

5. y = (2x)x3

6. y = xx2

7. y = x1/x

8. y = xlog x

5—8. Logarithmic Differentiation. When differentiating logarithmic and exponential functions, it is often more convenient to transform the given expression by making use of the properties of logarithms, namely:

log AB = log A + log B;log An = n log A; This is known as logarithmic differentiation, and is now illustrated.

EXAMPLE 1.Differentiate Solution.First write

y = log(x3 + 2),

Then EXAMPLE 2.Differentiate Solution.First write

y = log x2 − log (x + 1). EXAMPLE 3.Differentiate Solution.First write

y = [log (x2a) − log (x2 + a)].

Then EXAMPLE 4.Differentiate Solution.First take the logarithm of both sides:

log y = log (x + 3) + log (x + 2) − log (x + 1).

Then, differentiating both sides with respect to x: EXAMPLE 5.Differentiate y = xx+1.

Solution.First take the logarithm of both sides:

log y = (x+ 1) log x.

Differentiating both sides with respect to x: EXAMPLE 6.Differentiate Solution.Taking the logarithm of both sides:

log y = x2 log x.

Differentiating both sides with respect to x: · = x2 · + log x·2x

= x(1 + 2 log x). = xx2+1(1 + 2 log x).

EXERCISE 5—4

Differentiate, using the method of logarithmic differentiation: 2. y = xxn

3. y = exx  ﻿