Power Rule of Derivatives

The power rule of differentiation says that if the given function is of the form $x^n$, where $n$ is any constant, then we can differentiate the function in the following way:

Basics of Differentiation

This section covers key differentiation concepts, rules, and applications.

• Differentiability
• Derivative
• Derivative by First Principle
• Algebra of Derivative of Functions

Rules & Techniques of Differentiation

This section introduces the fundamental rules and methods used to differentiate functions efficiently.

• Rules for Differentiation
• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule
• Formulas for Differentiation
• Implicit Differentiation
• Logarithmic Differentiation
• Parametric Differentiation

$$f(x) = x^n$$ $$f'(x) = \frac{d}{dx}(x^n)$$ $$f'(x) = nx^{n-1}$$

This means that in such a case the differentiation is equal to the variable raised to 1 less than the original power and multiplied by the original power. Or, simply the power will be dropped in front of the variable (i.e. $x$ in this case) and the power is reduced by one.

Example: Differentiate the function $f(x) = x^3$ with respect to $x$.

Given:

$$f(x) = x^3$$ $$\Rightarrow f'(x) = \frac{d}{dx}(x^3)$$ $$\Rightarrow f'(x) = 3x^{3-1}$$ $$\Rightarrow f'(x) = 3x^2$$

Standard Derivative Rules Table

Example 2: Complex Polynomial

Find the derivative of $g(x) = 4x^3 - 2x^2 + 5x - 7$.

Solution:

$$g'(x) = \frac{d}{dx}(4x^3) - \frac{d}{dx}(2x^2) + \frac{d}{dx}(5x) - \frac{d}{dx}(7)$$ $$g'(x) = 4(3x^2) - 2(2x) + 5(1) - 0$$ $$g'(x) = 12x^2 - 4x + 5$$

# Check with Python SymPy
import sympy as sp
x = sp.Symbol('x')
g = 4*x**3 - 2*x**2 + 5*x - 7
print(sp.diff(g, x))

Output:

12*x**2 - 4*x + 5