Differentiating Powers
The Power Rule
The power rule states that for a function $f(x) = ax^n$, where $a$ is a constant, the derivative is:
$f'(x) = anx^{n-1}$
This rule forms the basis for differentiating many types of functions, including polynomials and radicals.
NoteThe power rule works for any exponent, including negative integers, fractions, and zero. However, in the context of polynomials, we typically deal with non-negative integer exponents.
Applying the Power Rule
To apply the power rule, follow these steps:
- Multiply the coefficient by the exponent
- Reduce the exponent by 1
Let's differentiate $f(x) = 3x^4$:
- Multiply 3 by 4: $3 \cdot 4 = 12$
- Reduce the exponent by 1: $4 - 1 = 3$
Therefore, $f'(x) = 12x^3$
Differentiating Polynomials
When dealing with polynomials that have multiple terms, we apply the power rule to each term individually and then combine the results. This is possible due to the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.
For a polynomial function of the form:
$f(x) = ax^n + bx^{n-1} + cx^{n-2} + ... + kx + m$
We differentiate each term separately and then combine the results:
$f'(x) = anx^{n-1} + b(n-1)x^{n-2} + c(n-2)x^{n-3} + ... + k$
ExampleLet's differentiate $f(x) = 2x^3 - 5x^2 + 3x - 7$:
- $2x^3$ becomes $6x^2$
- $-5x^2$ becomes $-10x$
- $3x$ becomes $3$
- The constant term $-7$ disappears in the derivative
Combining these results, we get: $f'(x) = 6x^2 - 10x + 3$
TipWhen differentiating polynomials with many terms, it can be helpful to work through the terms systematically, perhaps from highest power to lowest, to ensure you don't miss any terms.
Special Cases
Differentiating Constants
When differentiating a constant term (which can be thought of as $x^0$), the result is always 0. This is because the derivative represents the rate of change, and constants do not change with respect to $x$.
You can also think of this as differentiating $f(x) = ax^0$. In this case, using the power rule results in $f'(x) = 0ax^{-1} = 0$.
ExampleThe derivative of $f(x) = 5$ is $f'(x) = 0$
Differentiating Linear Terms
For linear terms (terms with $x^1$), the derivative is simply the coefficient of $x$. This is because $x^1$ can be thought of as $1x^1$, and applying the power rule gives us $1 \cdot 1x^{1-1} = 1$.
ExampleThe derivative of $f(x) = 7x$ is $f'(x) = 7$
Negative Integer Exponents
The power rule also applies to negative integer exponents. However, it's important to remember that negative exponents in the derivative will result in a more negative exponent.
ExampleLet's differentiate $f(x) = 3x^{-2}$:
Applying the power rule: $f'(x) = 3 \cdot (-2)x^{-3} = -6x^{-3}$
Common MistakeStudents often forget to apply the negative sign when differentiating terms with negative exponents, or they mistakenly make the exponent less negative instead of more negative.
Non-integer powers
The power rule also applies to non-integer powers, such as $f(x) = x^{\frac12}$. Remember that the power is still subtracted by 1, which can sometimes make the power in the derivative negative.
ExampleLet's differentiate $f(x) = \sqrt[3]{x}$:
The cube root is really just a power of $\frac13$, so we can rewrite this as $f(x) = x^{\frac13}$.
Therefore, using the power rule, $f'(x) = \frac13 x^{-\frac23}$, or $f'(x) = \frac{1}{3\sqrt[3]{x^2}}$.
Practical Applications
Understanding how to differentiate polynomials is crucial in many areas of mathematics and science. Some practical applications include:
- Optimization problems in economics and engineering
- Calculating rates of change in physics
- Analyzing the behavior of polynomial functions
The ability to quickly and accurately differentiate polynomials is a fundamental skill that will be built upon in more advanced calculus topics.
