The Constant Rule (and Constant Multiple Rule)
Two of the simplest — yet most used — rules in differentiation are the Constant Rule and the Constant Multiple Rule. They let you handle constants in derivatives quickly and correctly.
📜 Constant Rule
The derivative of a constant is zero:
💡 A constant doesn’t change with $x$, so its slope is $0$ everywhere.
📜 Constant Multiple Rule
If a function is multiplied by a constant factor, the constant factors out of the derivative:
💡 The constant scales the function — and therefore its slope — by the same factor.
🪜 When and How to Apply
- Pure constants → 0: Any standalone number ($5$, $-3$, $\pi$) has derivative $0$.
- Factor out constants: For $c\,u(x)$, keep $c$ and differentiate only $u(x)$.
- Combine with other rules: Use together with sum, product, quotient, or chain rule.
✏️ Examples
Example 1: Pure Constant
Given: $f(x) = 7$
Derivative: $$ f'(x) = 0. $$
Example 2: Constant Multiple of a Power
Given: $f(x) = 4x^3$
Derivative: $$ \frac{d}{dx}(4x^3) = 4 \cdot \frac{d}{dx}(x^3) = 4\cdot 3x^2 = 12x^2. $$
Example 3: Sum with a Constant Term
Given: $f(x) = x^2 - 5x + 9$
Derivative: $$ f'(x) = 2x - 5. \quad (\frac{d}{dx}(9)=0) $$
Example 4: Constant Multiple with Trig
Given: $f(x) = -3\sin x$
Derivative: $$ f'(x) = -3\cos x. $$
Example 5: Constant Multiple + Chain Rule
Given: $f(x) = \tfrac{1}{2}(2x+1)^4$
Derivative: $$ f'(x) = \tfrac{1}{2}\cdot 4(2x+1)^3 \cdot 2 = 4(2x+1)^3. $$