The Calculus Primer (2011)

Part V. Differentiation of Transcendental Functions

Chapter 17. DERIVATIVES OF EXPONENTIAL FUNCTIONS

5—6. Differentiating the Simple Exponential Function. Let us consider first the simple exponential function y = a^v, 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 [2]:

Now we apply the formula for the derivative of a function of a function, §5—1, equation [1], 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 (a^4x+1).

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

EXAMPLE 2.Differentiate y = e^−2x.

Solution. = e^−2x(−2) = −2e^−2x.

EXAMPLE 3.Differentiate s = e^t3+1.

Solution.

EXAMPLE 4.Find in the equation ρ = a^θ2.

Solution. = log a·a^θ2·2θ.

EXAMPLE 5.Differentiate y = e^{log x}.

Solution.

But e^{log x} = x, by the definition of a logarithm.

Therefore

EXAMPLE 6.Differentiate y = xe^2/x.

Solution.

EXERCISE 5—2

Differentiate:

1. y = k^4x

3. y = e^kx

4. y = ke^x

5. y = e^5−3t

11. ρ = a^θ

12. ρ = a^logθ

13. ρ = e^aθ

14. y = e^1/x

15. y = xe^x

16. y = x²e^2x

17. y = xⁿ + n^x

18. y = a^xx^a

19. y = e^x(x² − 2x + 2)

20. y = e^x(x − 1)

5—7. Differentiating the General Exponential Function. We now consider the more general exponential function y = u^v, 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 = u^v.(1)

Take the logarithm of both sides to the base e:

log_e y = v log_e u,

or, by the definition of a logarithm,

y = e^{v log u}.(2)

Differentiating equation (2) by formula [4a]:

But e^{v log u} = y from equation (2); and y = u^v from equation (1); hence e^{v log u} = u^v.

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 = x^x.

Solution.

EXAMPLE 2.Differentiate

Solution.Here u = x, v = e^x.

EXAMPLE 3.Prove that the formula holds for all values of the constant n by setting v = n in formula [5].

Solution.If v = n, we have, from [5]:

EXERCISE 5—3

Differentiate:

1. y = x^x+1

2. y = (x² + 1)^x

3. y = (x³)^−x

4. y = x^2x

5. y = (2x)^x3

6. y = x^x2

7. y = x^1/x

8. y = x^{log 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),

thus eliminating the radical.

Then

EXAMPLE 2.Differentiate

Solution.First write

y = log x² − log (x + 1).

EXAMPLE 3.Differentiate

Solution.First write

y = [log (x² − a) − 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 = x^x+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).

= x^x2+1(1 + 2 log x).

EXERCISE 5—4

Differentiate, using the method of logarithmic differentiation:

2. y = x^xn

3. y = e^xx