Constant Multiple Rule of Derivatives

Neetesh Kumar | February 08, 2025 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition of Constant Multiple Rule
• 2. Explanation & Intuition
• 3. Solved Examples
• 4. Conclusion

Constant Multiple Rule:
If $c$ is a constant and $f(x)$ is a differentiable function, then:

$\boxed{\dfrac{d}{dx} \big[ c \cdot f(x) \big] = c \cdot \dfrac{d}{dx} \big[f(x) \big]}$

This means that when differentiating a function multiplied by a constant, we simply multiply the derivative of the function by that constant.

Example 1: Differentiating a Quadratic Function

Differentiate $f(x) = 7x^3$.

Solution:

Using the constant multiple rule:

$f'(x) = \dfrac{d}{dx} \big( 7x^3 \big) = 7 \cdot \dfrac{d}{dx} \big( x^3 \big)$

Since $\dfrac{d}{dx} (x^n) = n x^{n-1}$, we get:

$f'(x) = 7 \cdot 3x^2 = \boxed{21x^2}$.

Example 2: Differentiating a Trigonometric Function

Differentiate $f(x) = 5\sin x$.

Solution:

Using the constant multiple rule:

$f'(x) = 5 \cdot \dfrac{d}{dx} (\sin x)$.

Since $\dfrac{d}{dx} (\sin x) = \cos x$, we get:

$f'(x) = 5 \cos x$.

Thus, the final answer is:

$f'(x) = \boxed{5\cos x}$.

Example 3: Differentiating a Logarithmic Function

Find the derivative of $f(x) = 5\ln(x)$.

Solution:

Using the constant multiple rule:

$f'(x) = 5 \cdot \dfrac{d}{dx} (\ln x)$.

Since $\dfrac{d}{dx} (\ln x) = \dfrac{1}{x}$, we get:

$f'(x) = (5).\bigg(\dfrac{1}{x}\bigg)$.

Thus, the final answer is:

$f'(x) = \boxed{\dfrac{5}{x}}$.