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.
