## 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) = −2*e*^{−2x}.

EXAMPLE 3.Differentiate *s* = *e ^{t}*

^{3}

^{+1}.

*Solution*.

EXAMPLE 4.Find in the equation *ρ* = *a ^{θ}*

^{2}.

*Solution*. = log *a*·*a ^{θ}*

^{2}·2

*θ*.

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* = *x*^{2}*e*^{2x}

**17.** *y* = *x ^{n}* +

*n*

^{x}**18.** *y* = *a ^{x}x^{a}*

**19.** *y* = *e ^{x}*(

*x*

^{2}− 2

*x*+ 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.

Let*y* = *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*from equation (1); hence

^{v}*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* = (*x*^{2} + 1)^{x}

**3.** *y* = (*x*^{3})^{−x}

**4.** *y* = *x*^{2x}

**5.** *y* = (2*x*)^{x}^{3}

**6.** *y* = *x ^{x}*

^{2}

**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 *A ^{n}* =

*n*log

*A*;

This is known as *logarithmic differentiation*, and is now illustrated.

EXAMPLE 1.Differentiate

*Solution*.First write

*y* = log(*x*^{3} + 2),

thus eliminating the radical.

Then

EXAMPLE 2.Differentiate

*Solution*.First write

*y* = log *x*^{2} − log (*x* + 1).

EXAMPLE 3.Differentiate

*Solution*.First write

*y* = [log (*x*^{2} − *a*) − log (*x*^{2} + *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* = *x*^{2} log *x.*

Differentiating both sides with respect to *x*:

· = *x*^{2} · + log *x*·2*x*

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

= *x ^{x}*

^{2}

^{+1}(1 + 2 log

*x*).

**EXERCISE 5—4**

*Differentiate*, *using the method of logarithmic differentiation:*

**2.** *y* = *x ^{x}*

^{n}

**3.** *y* = *e ^{x}*

^{x}