Derivative Calculator (Power Rule)
Calculate derivatives using the power rule: d/dx[ax^n] = an·x^(n-1). Find derivatives of polynomial functions with step-by-step solutions.
∫ Calculus📐 d/dx[axⁿ] = an·xⁿ⁻¹🔢 Math
Coefficient a
Exponent n
Evaluate at x
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Derivative Calculator (Power Rule) | — | d/dx[axⁿ] = an·xⁿ⁻¹ | expression |
Step-by-Step Examples
Example 1
3x²
d/dx[3x²]
- nc=3×2=6, ne=2-1=1
- = 6x
✓ 6x
Example 2
5x⁴
d/dx[5x⁴]
- nc=20, ne=3
- = 20x³; at x=2: 160
✓ 20x³
Example 3
Constants
d/dx[9]
- Constant rule: d/dx[c]=0
- d/dx[4x]=4
✓ d/dx[9]=0
Real-World Applications
Physics
Velocity = derivative of position. Acceleration = derivative of velocity.
Economics
Marginal cost = derivative of total cost.
Optimization
f'(x)=0 gives maxima and minima.
Engineering
Rates of change in circuits and fluid dynamics.
Common Mistakes to Avoid
⚠️
Missing exponent reduction
d/dx[x³]=3x², NOT 3x³. Both multiply AND reduce exponent.
⚠️
Constants become zero
d/dx[5]=0. Constants have no rate of change.
⚠️
Power rule only for x^n
Use different rules for sin(x), e^x, ln(x).
Frequently Asked Questions
What is the power rule? ▾
d/dx[ax^n]=a·n·x^(n-1). Works for any real n.
What is d/dx[x]? ▾
d/dx[x¹]=1.
What does derivative mean? ▾
Instantaneous rate of change; slope of the tangent line.
Sum rule? ▾
Differentiate each term separately: d/dx[3x²+5x]=6x+5.
d/dx[√x]? ▾
d/dx[x^(1/2)]=(1/2)x^(-1/2)=1/(2√x).
Related Math Calculators
Formula Explorer connections
Interpretation: the exponent becomes the new coefficient and then decreases by one. The result gives the instantaneous rate of change of axⁿ. Assumption: the expression must be differentiable at the evaluation point; fractional or negative powers may restrict the real-number domain.