Calculus I · Unit 2A · lesson

Sums and Constant Multiples

Concept

Learning objectives

Use constant-multiple and sum rules; differentiate polynomial and mixed power expressions efficiently.

Linearity of Differentiation

Explanation

Before the formulas

The shortcuts in Sums and Constant Multiples do not replace the limit definition. They package conclusions already justified from it. That distinction matters because a rule applies only when the function has the relevant structure and lies in its domain. A familiar-looking exponent or fraction is not permission to differentiate blindly.

After using a rule, check the result structurally. A derivative of a polynomial should have lower degree. A product derivative should usually contain contributions from both factors. A quotient result should preserve the denominator restriction. These checks are quick enough to use on homework and valuable enough to use on exams.

Explanation

Differentiate a sum piece by piece

If an output is built by adding several contributions, a small change in the total is the sum of the small changes in those contributions. That is why derivatives distribute across sums and differences. A constant multiplier simply scales every output change by the same amount, so it scales the derivative too.

This linear behavior is one of the cleanest parts of differentiation. It lets a long polynomial be handled term by term rather than as one giant object. Keep the signs attached to their terms and simplify only after each derivative is correct.

Many models are built by adding simpler effects: fixed cost plus variable cost, baseline temperature plus oscillation, or several forces acting together. Differentiation respects that addition. Each component contributes its own rate, and a constant multiplier scales the rate by the same amount.

This linearity is why complicated-looking polynomials are routine. It is also why units and interpretation can be handled component by component, provided every term describes the same output quantity.

If $c$ is constant, then

\frac{d}{dx}[cf(x)]=cf'(x).

If $f$ and $g$ are differentiable, then

\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x).

Together these say differentiation is a linear operation:

\frac{d}{dx}[af(x)+bg(x)]=af'(x)+bg'(x).

Guided walkthrough

A mixed power expression

Differentiate

y=3x^{7/2}-4x^{-2}+9.

Answer reveal

Worked solution

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Proof idea

The sum rule follows directly from limits because

\frac{(f+g)(x+h)-(f+g)(x)}{h} =\frac{f(x+h)-f(x)}h+\frac{g(x+h)-g(x)}h.

Taking limits separates the sum into $f'(x)+g'(x)$ .

Interactive checklinearity-01

Differentiate $2x^5-2x^{-3}+8$ .

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Show hint

Use the power rule on both variable terms; the constant contributes zero.

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Application

Total electrical load

Suppose three devices draw power

P_1(t)=40+2t, \quad P_2(t)=25+3t^2, \quad P_3(t)=10e^{0.1t}.

The total load is $P=P_1+P_2+P_3$ , so

P'(t)=2+6t+e^{0.1t}.

Each device contributes independently to the rate at which the building's total power demand changes. This is linearity in a context where the units remain watts per unit time throughout.

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?

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