Differentiating a function involves finding its derivative, which represents its rate of change. For a power function, the differentiation process is relatively straightforward, involving a specific rule. Let's clarify how to approach this.
Differentiating a Power Function
A power function is a function of the form f(x) = xn, where n is a constant. The power rule of differentiation provides a simple method for finding its derivative.
The Power Rule
According to the reference, the core principle is: bringing the exponent value down and multiplying it to the function and then subtracting one from the exponent.
This translates to:
If f(x) = xn, then f'(x) = nx(n-1)*
Example
Let's say we have the function f(x) = x3. To find its derivative:
- Bring the exponent (3) down and multiply it by the function: 3x3
- Subtract 1 from the exponent: 3x(3-1) = 3x2
Therefore, the derivative of f(x) = x3 is f'(x) = 3x2.
Generalizing to Functions within a Power
The power rule can be extended to functions raised to a power. Let's say we have f(x) = [g(x)]n. The chain rule combined with the power rule tells us:
f'(x) = n[g(x)](n-1) g'(x)*
Here, g'(x) represents the derivative of the inner function g(x).
Example
Let's differentiate f(x) = (x2 + 1)4.
- Identify the outer power function: u4 where u = x2 + 1
- Apply the power rule and the chain rule: f'(x) = 4(x2 + 1)3 (2x)*
- Simplify: f'(x) = 8x(x2 + 1)3
Summary
| Function Type | Differentiation Rule | Example |
|---|---|---|
| Power Function: f(x) = xn | f'(x) = nx(n-1)* | f(x) = x5 => f'(x) = 5x4 |
| Function to a Power: f(x) = [g(x)]n | f'(x) = n[g(x)](n-1) g'(x)* | f(x) = (sin(x))2 => f'(x) = 2sin(x)cos(x) |
