Constant Multiple Rule
Constants factor out of a derivative.
Formula
Variables
Example
d/dx (5x³) = 5·3x² = 15x².
How it works
If a function is multiplied by a constant, that constant stays unchanged when you differentiate. This works because differentiation measures how the function changes, and a fixed number like 3 or -1/2 does not affect the rate of change except by scaling it. In other words, the derivative of the scaled function is the same scale factor times the derivative of the original function.
When to use it
Use this whenever the function you are differentiating has a constant multiplier in front, such as 7sin(x), -4x^3, or 1/2 e^x. It is often the first step before applying other rules like the power rule, product rule, or chain rule.
Worked example, step by step
Differentiate f(x) = 8x^4 Step 1: Identify the constant multiplier: 8 Step 2: Differentiate the variable part: d/dx(x^4) = 4x^3 Step 3: Multiply the derivative by the constant: 8 · 4x^3 Step 4: Simplify: f'(x) = 32x^3
Common mistakes to avoid
- Differentiating the constant multiplier itself instead of leaving it unchanged
- Forgetting to multiply the final derivative by the constant factor
- Applying the rule only to numbers and not to expressions like -3(x^2+1)
Did You Know?
A constant factor just rides along through differentiation, unchanged.
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