Power Rule Calculus Calculator
Calculate the derivative and indefinite integral for functions in the form of f(x) = axⁿ
How to Use This Calculus Calculator
This tool is specifically designed to handle the Power Rule, which is the foundation of differential and integral calculus. It solves equations where a variable (x) is raised to a constant power (n) and multiplied by a coefficient (a).
Understanding the Derivative Power Rule
In calculus, the Power Rule for differentiation states that for any real number n, the derivative of xⁿ is nxⁿ⁻¹. If there is a coefficient a, the formula becomes:
For example, if you have 5x³, the derivative is 5 * 3 * x^(3-1), which simplifies to 15x².
Understanding the Integral Power Rule
The Power Rule for integration (the anti-derivative) is the inverse. To find the integral of axⁿ, you increase the exponent by 1 and divide the coefficient by the new exponent. The formula is:
Note: This rule applies for all values of n except for n = -1. When n = -1, the integral is a * ln|x| + C.
Real-World Example
If you are calculating the velocity of an object where the position is defined by 4x²:
- Coefficient (a): 4
- Exponent (n): 2
- Derivative (Velocity): 8x¹ (or simply 8x)
- Integral: (4/3)x³ + C (approximately 1.33x³)