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 are the backbone for quickly handling constants in derivatives.
📜 Constant Rule
The derivative of a constant is zero:
💡 Why? A constant function does not change with $x$, so its slope is zero everywhere.
📜 Constant Multiple Rule
If a function is multiplied by a constant factor, the constant can be pulled in front of the derivative:
💡 Interpretation: The constant scales the function and therefore scales its slope by the same factor.
🪜 When and How to Apply
- Check for pure constants: Any standalone number (e.g., $5$, $-3$, $\pi$) has derivative $0$.
- Pull out constant factors: If your term is $c\cdot u(x)$, keep $c$ and differentiate $u(x)$ only.
- Combine with other rules: Use together with the sum, product, quotient, or chain rule as needed.
✏️ Examples for Illustration
Example 1: Pure Constant
Given: $f(x) = 7$
Derivative: $$ f'(x) = 0. $$
Example 2: Constant Multiple of a Power
Given: $f(x) = 4x^3$
- Pull out the constant: $\dfrac{d}{dx}(4x^3) = 4 \cdot \dfrac{d}{dx}(x^3) = 4 \cdot 3x^2$
Derivative: $$ f'(x) = 12x^2. $$
Example 3: Sum with a Constant Term
Given: $f(x) = x^2 - 5x + 9$
- $\dfrac{d}{dx}(x^2) = 2x$, $\dfrac{d}{dx}(-5x) = -5$, $\dfrac{d}{dx}(9) = 0$
Derivative: $$ f'(x) = 2x - 5. $$
Example 4: Constant Multiple with Trig
Given: $f(x) = -3\sin(x)$
- $\dfrac{d}{dx}\big(-3\sin x\big) = -3\cos x$
Derivative: $$ f'(x) = -3\cos x. $$
Example 5: Constant Multiple + Chain Rule
Given: $f(x) = \dfrac{1}{2}\,(2x+1)^4$
- Pull out $\tfrac{1}{2}$ and use chain rule on $(2x+1)^4$: $$ f'(x) = \tfrac{1}{2}\cdot 4(2x+1)^3 \cdot \frac{d}{dx}(2x+1) = \tfrac{1}{2}\cdot 4(2x+1)^3 \cdot 2 = 4(2x+1)^3. $$
Derivative: $$ f'(x) = 4(2x+1)^3. $$