Derivative Formulas for Powers and Polynomials

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. For powers and polynomials, there are specific formulas and rules that make finding derivatives more straightforward.

The Power Rule

The power rule is one of the most basic and important derivative formulas. It states that:

This rule applies to any real number exponent n, including whole numbers, fractions, negative numbers, and irrational numbers.

Applications of the Power Rule

Constant Multiple Rule

When a constant multiplies a power function, the constant factor can be factored out of the derivative:

Sum and Difference Rules

The derivative of a sum or difference of functions is the sum or difference of their derivatives:

Derivatives of Polynomials

A polynomial function is of the form P(x) = ax + ax + ... + ax + a, where a, a, ..., a, a are constants and n is a non-negative integer.

To find the derivative of a polynomial, we apply the power rule, constant multiple rule, and sum/difference rules to each term:

Special Cases

Derivative of a Constant Function

Derivative of the Identity Function

Derivative of Linear Functions

Common Powers and Their Derivatives

Function f(x)	Derivative f'(x)
x	2x
x	3x
x	4x
1/x = x	-1/x = -x
x = x/	1/(2x) = (1/2)x/
1/x = x/	-1/(2xx) = -(1/2)x/

The Product Rule for Powers

When finding the derivative of a product of power functions, we can use the product rule:

Alternatively, we can simplify first using the exponent rule x^m x^n = x^(m+n) and then apply the power rule.

The Quotient Rule for Powers

For the derivative of a quotient of power functions:

Again, we could first simplify using the quotient rule for exponents: x^m/x^n = x^(m-n), and then apply the power rule.

The Chain Rule with Powers

When a power function contains another function inside it, we use the chain rule:

Applications of the Chain Rule

Derivatives of Polynomial Functions with Chain Rule

For polynomial functions where a polynomial is raised to a power, we combine the power rule with the chain rule:

Applications of Polynomial Derivatives

Finding Critical Points

To find critical points of a polynomial function P(x), we find where P'(x) = 0 or P'(x) is undefined. Since polynomial derivatives are always defined, we only need to solve P'(x) = 0.

Determining Increasing and Decreasing Intervals

Polynomial derivatives help determine where a polynomial function is increasing or decreasing:

• If P'(x) > 0 on an interval, then P(x) is increasing on that interval
• If P'(x) < 0 on an interval, then P(x) is decreasing on that interval

Finding Local Extrema

Polynomial derivatives are instrumental in identifying local maxima and minima:

1. Find critical points by setting P'(x) = 0
2. Apply the first derivative test or second derivative test to classify each critical point

Concavity and Inflection Points

The second derivative P''(x) of a polynomial helps determine concavity:

• If P''(x) > 0 on an interval, then P(x) is concave up on that interval
• If P''(x) < 0 on an interval, then P(x) is concave down on that interval
• Inflection points occur where P''(x) = 0 and the concavity changes

Higher-Order Derivatives of Polynomials

We can apply the power rule repeatedly to find higher-order derivatives:

Summary of Derivative Rules for Powers and Polynomials

1. Power Rule: If f(x) = x, then f'(x) = nx
2. Constant Multiple Rule: If f(x) = kx, then f'(x) = knx
3. Sum/Difference Rule: The derivative of a sum/difference is the sum/difference of derivatives
4. Constant Function: If f(x) = c, then f'(x) = 0
5. Identity Function: If f(x) = x, then f'(x) = 1
6. Linear Function: If f(x) = mx + b, then f'(x) = m
7. Chain Rule: If f(x) = [g(x)], then f'(x) = n[g(x)] g'(x)

These formulas and rules form the foundation for differentiating polynomial functions of all types and are essential tools in calculus for solving a wide range of problems in mathematics, physics, economics, and other sciences.