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Calculus I · Unit 2A · lesson

The Constant and Power Rules

Concept

Learning objectives

Differentiate constants and power functions; use constant multiples, sums, and differences; distinguish a coefficient from an exponent.

The Core Differentiation Rules

Constants, Powers, Sums, and Differences

Explanation

Before the formulas

In The Constant and Power Rules, the challenge is not remembering a longer formula list. It is seeing how an expression was built. Differentiation rules mirror construction rules: sums are handled term by term, products require two contributions, quotients balance numerator and denominator change, and compositions require the chain rule. The outermost operation determines the first move.

Write a one-line rule plan before doing algebra. For example, label an expression "product outside, chain rule in the second factor." This takes seconds and prevents the most expensive errors. Keep grouping visible until the derivative structure is complete; simplify only when simplification makes the next decision clearer.

For x n, differentiation multiplies by the old exponent and lowers the exponent by one. The paired graphs show how the derivative records the original curve's changing steepness.
Read this graph as text

The power rule changes both exponent and slope pattern. For x n , differentiation multiplies by the old exponent and lowers the exponent by one. The paired graphs show how the derivative records the original curve's changing steepness. For x 2 , slopes are negative on the left, zero at the origin, and positive on the right, so the derivative is the line 2x . For x 3 , the graph increases everywhere and flattens at the origin, so the derivative 3x 2 is nonnegative and touches zero there. The rule matches the geometry rather than merely manipulating symbols.

The visual uses labeled positions, solid and dashed line styles, and written descriptions so the power rule changes both exponent and slope pattern does not depend on color.

Why it matters: This four-panel visual should make the power rule memorable through shape correspondence. It also corrects the misconception that a zero derivative always means a local maximum or minimum: x 3 has derivative zero at the origin but continues increasing.

Visual study

For x n, differentiation multiplies by the old exponent and lowers the exponent by one. The paired graphs show how the derivative records the original curve's changing steepness.

Explanation

The exponent moves because the local change has one fewer factor

The power rule can look like a trick: bring the exponent down and subtract one. Its origin is less magical. Expanding (x+h)n(x+h)^n produces a leading change term nxn−1hnx^{n-1}h; after division by hh, the coefficient nxn−1nx^{n-1} remains while higher powers of hh vanish in the limit.

When using the rule, rewrite radicals and reciprocals as powers first. That single habit turns many apparently different formulas into the same pattern and makes sign errors easier to catch.

The power rule compresses a limit calculation into a two-move pattern: multiply by the exponent and lower that exponent by one. The pattern is simple, but it represents a profound fact about how scaling laws respond to change. Area grows like length squared, volume like length cubed, and their derivatives reveal how quickly those quantities become more sensitive as size increases.

Use the rule term by term, and keep coefficients separate from exponents. The derivative operator is linear, so a polynomial can be dismantled, differentiated in pieces, and reassembled without changing the result.

The limit definition explains what a derivative means. Differentiation rules make routine computation efficient. The rules are not replacements for the definition; they are results proved from it.

Theorem

Constant and power rules

For a constant cc,

ddx[c]=0.\frac{d}{dx}[c]=0.

For any real exponent nn on an interval where xnx^n is defined and differentiable,

ddx[xn]=nxn−1.\boxed{\frac{d}{dx}[x^n]=nx^{n-1}}.

A constant function has a horizontal graph, so its slope is zero. The power rule performs two visible moves: multiply by the old exponent, then reduce the exponent by one.

Guided walkthrough

Use the power rule one term at a time

Differentiate

f(x)=4x5−3x2+7x−9.f(x)=4x^5-3x^2+7x-9.
Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Proof idea

Why the integer power rule has this shape

For f(x)=xnf(x)=x^n,

(x+h)n−xnh.\frac{(x+h)^n-x^n}{h}.

The binomial expansion begins

(x+h)n=xn+nxn−1h+(n2)xn−2h2+⋯+hn.(x+h)^n=x^n+nx^{n-1}h+\binom n2x^{n-2}h^2+\cdots+h^n.

After subtracting xnx^n and dividing by hh, the first surviving term is nxn−1nx^{n-1}; every other term still contains a positive power of hh and vanishes as h→0h\to0.

Common mistake

For 5x35x^3, the derivative is 15x215x^2, not 5x25x^2 and not 3x23x^2. The coefficient stays and the exponent comes down as an additional factor.

Interactive checkpower-rule-01

Differentiate 3x6−2x4+2x−113x^6-2x^4+2x-11.

Your work stays on this device. No account or AI grader is used.

Show hint

Differentiate each power term and remember that constants disappear.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Application

Why larger spheres gain volume so quickly

The volume of a sphere is V(r)=43πr3V(r)=\frac43\pi r^3. Differentiating with respect to radius gives

V′(r)=4πr2.V'(r)=4\pi r^2.

At r=2r=2, adding a small 0.010.01 unit to the radius increases volume by approximately

V′(2)(0.01)=16π(0.01)≈0.503.V'(2)(0.01)=16\pi(0.01)\approx0.503.

At r=10r=10, the same radial increase produces about 4π4\pi cubic units. Sensitivity grows with surface area, which is not a coincidence: a thin outer shell has volume approximately surface area times thickness.

Optional advanced note

Beyond integer powers

The binomial proof directly handles positive integers. Extending the power rule to negative integers, rational exponents, and arbitrary real exponents requires additional arguments and careful domains. One route writes xr=erln⁡xx^r=e^{r\ln x} for x>0x>0 and uses the chain rule. The familiar formula survives, but its proof depends on the theory of exponential and logarithmic functions.

After the explanation

Use the section idea

Reading lens

Every rule is a compressed limit calculation; choose the structure before doing algebra.

Mental model

Sums contribute independently, products have two changing contributions, and quotients must account for a changing denominator.

Decision

Name the outermost algebraic structure, then apply the smallest rule set that preserves it.

Common trap

Applying a familiar rule to the wrong outer structure or simplifying after a differentiation error.

Check yourself

Can you justify the primary rule before writing the first derivative symbol?

Interactive checkpower-extra-01

Differentiate 4x54x^5.

Your work stays on this device. No account or AI grader is used.

Show hint

Bring down the exponent and reduce it by one.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

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