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DERIVATIVE OF CONSTANTS TO VARIABLE POWERS In this section two forms of a constant to a variable
power will be presented. The two exponential functions will be
and
, where x
is the variable, a is any constant, and e is equal to
2.71828.... Recalling our study of logarithms in Mathematics,
Volume 2A, since In and e are inverse functions, then
and
If
then
PROOF: Since y = In x is differentiable, so is its
inverse, y =
. To
obtain the derivative of y =
, we
differentiate both sides of equation (5.10) with respect to x, which gives
Multiplying both sides of equation (5.12) by
gives
Chain rule differentiation and equation (5.11) give
SOLUTION.
SOLUTION.
If _{ } then
PROOF. Applying logarithmic rules, _{
} Differentiating both sides of equation (5.14) gives _{}
NOTE: In a is a constant.
SOLUTION:
Chain rule differentiation and equation (5.13) give
^{} SOLUTION:
PRACTICE
PROBLEMS: Find d of the following:
ANSWERS:
SUMMARY The following are the major topics covered in this
chapter: 1. Derivative of a constant: Theorem 1. The derivative of a constant is zero.
2. Derivative of a variable raised to a power: Theorem 2. The derivative of the function
is given by
, if n is
any real number. 3. Derivative of the sum of two or more functions: Theorem 3. The derivative of the sum of two or more
differentiable functions of x is equal to the sum of their derivatives. If two functions of x are given, such that u = g(x) and v = h(x), and also y = u + v = g(x) + h(x), then
4. Derivative of the product of two or more functions: Theorem 4. The derivative of the product of two differentiable
functions of x is equal to the first function multiplied by the derivative of
the second function, plus the second function multiplied by the derivative of
the first function.
This theorem can be extended to three or more functions.
5. Derivative of the quotient of two functions: Theorem 5. At a point where the denominator
is not equal to zero, the derivative of the quotient of two differentiable
functions of x is equal to the denominator times the derivative of the
numerator minus the numerator times the derivative of the denominator, all
divided by the square of the denominator.
6. Derivative of a function raised to a power: Theorem 6. The derivative of any
differentiable function of x raised to the power n, where n is any real number,
is equal to n times the polynomial function of x to the (n  ])power times the
derivative of the polynomial itself. If y =
, where u
is any differentiable function of x, then
7. Derivative of a function in radical form: To
differentiate a function containing a radical, replace the radical by a fractional
exponent; then find the derivative by applying the appropriate theorems. 8. Derivative of a function using the chain rule:
where the variable y = f(u) is a differentiable function
of u and u = g(x) is differentiable function of x. 9. Derivative of an inverse function: Theorem 7. The derivative of an inverse
function is equal to the reciprocal of the derivative of the direct function.
10. Derivative of an implicit function: In equations
containing x and y, if an equation of y is not solved for, then y is called an implicit
function of x. The derivative of each term containing y will be
followed by
11. Derivative of trigonometric functions:
12. Derivative of natural logarithmic functions: Theorem 8. The natural logarithm y = In x has
the derivative _{
} If u is a positive differentiable function of x, then
13. Derivative of a constant to a variable power:
where x is a variable, u is a function of x, a is a
constant, and e is equal to 2.71828.... 