Power & Sum Rules
Measure local slopes of power functions, identify the reusable pattern, then differentiate polynomial terms without rebuilding the limit definition from scratch each time.
At x = 1, tiny secants on x², x³, and x⁴ approach slopes 2, 3, and 4.
The exponent seems to be showing up inside the slope. Is that only a coincidence at x = 1, or can one algebraic expression reproduce the local slopes everywhere?
Measure first. Guess the derivative form second.
Choose a power function, squeeze a symmetric secant around one point, and watch its slope settle. Then test whether one candidate expression a·xᵇ reproduces the near-tangent slopes at several x-values.
at x = 1.15 using points 0.45 units to either side. Shrink the width to make the local measurement more tangent-like.
Try x = 1 first for several powers. Then move away from 1. Which part of your candidate seems to come from the old exponent, and which part seems to control how slope changes with x?
x − h
symmetric secant
x + h
The derivative of a power is another power with one less degree.
For the polynomial powers explored in the sandbox, the slope measurements fit a compact rule: multiply by the old exponent, then reduce that exponent by one. This is the power rule.
The rule is a shortcut built on derivative structure, not a replacement for what a derivative means. It produces the function whose value gives the original function's local rate of change.
This lesson focuses on positive-integer powers used in polynomials. Broader versions of the power rule require attention to domains and how powers are defined.
Polynomials become manageable because differentiation respects scaling and addition.
For the positive-integer polynomial powers explored here, the old exponent becomes a coefficient and the exponent decreases by one.
A fixed multiplier scales every local rate by the same factor.
Addition and subtraction let derivative work distribute term by term.
A constant function does not change as x changes, so its local rate is zero.
Start with the original polynomial.
Use the power rule on the power terms, d/dx(x)=1 on the linear term, and d/dx(constant)=0.
The original coefficients stay as multipliers.
The derivative is already assembled term by term; now only arithmetic remains.
This new function returns the local slope of f at each x.
The old exponent does two jobs: it becomes a coefficient and the remaining power drops by one.
If terms are added or subtracted, differentiate each term. If two changing functions are multiplied, is not generally . That is why the Product Rule needs its own lesson.