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Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Lesson ~10 min read

In simple terms: In simple terms, this lesson teaches you easy shortcuts—the Constant, Sum, Difference, and Constant Multiple rules—to find the derivative of any polynomial function piece by piece.

Why this matters

Imagine you're building a custom gaming PC for your friend, Marcus. You have all the parts laid out: the CPU, the graphics card, the RAM, the motherboard. It looks overwhelming all at once. But you don't assemble it all at once, do you? You handle each component one by one. You install the CPU, then you click in the RAM, then you seat the graphics card. You have a specific, simple process for each part.

Finding derivatives of complex functions can feel the same way. Looking at a long polynomial like f(x) = 5x⁴ - 9x³ + x² - 20 can feel like that pile of PC parts. In this lesson, we're going to learn the simple, step-by-step rules that let you handle each piece of the function separately, making the whole process manageable and fast.

Concept overview

flowchart TD
    A[Start with a polynomial function f(x)] --> B{Break f(x) into individual terms};
    B --> C{For each term 'ax^n'};
    C --> D{Is it a constant? (n=0)};
    D -- Yes --> E[Derivative is 0];
    D -- No --> F[Apply Constant Multiple Rule];
    F --> G[Keep 'a', find derivative of x^n];
    G --> H[Apply Power Rule: n*x^(n-1)];
    H --> I[Multiply: a * n*x^(n-1)];
    E --> J[Combine all term derivatives];
    I --> J;
    J --> K[Final derivative f'(x)];

Core explanation

So far, you've used the limit definition of the derivative. It's powerful, and it's the bedrock of calculus, but let's be honest: it's slow. It's like using a hand-cranked drill when you have a power tool sitting right there. Today, we're plugging in the power tools.

These rules—the Constant, Constant Multiple, Sum, and Difference Rules—work together with the Power Rule you've already seen. They allow us to take apart a complicated-looking function, deal with each piece simply, and then put it all back together.

The Constant Rule

Let's start with the easiest one. What's the rate of change of something that... doesn't change?

The Rule: The derivative of any constant c is 0. d/dx [c] = 0

Think about the function f(x) = 4. This is a horizontal line passing through y=4. What's its slope at any point? It's zero. It's perfectly flat. The rate of change is zero.

If your friend Priya has a constant $50 in her savings account for a whole year, the rate of change of her money is $0 per day. It's not changing. That's all the Constant Rule is.

If f(x) = -12, then f'(x) = 0
If g(x) = π, then g'(x) = 0 (Yes, π is just a constant!)

The Constant Multiple Rule

Okay, what if a constant is multiplying a function?

The Rule: The derivative of a constant c times a function f(x) is the constant times the derivative of the function. d/dx [c * f(x)] = c * f'(x)

In simple terms, the constant just comes along for the ride. You can pull it out, focus on differentiating the function part, and then multiply the constant back in at the end.

Imagine you have a function f(x) that represents the incline of a hiking trail. Now consider g(x) = 3 * f(x). This new trail is just a scaled-up version of the original; at every point, it's exactly 3 times as steep. The 3 just scales the steepness (the derivative).

Let's see it with the Power Rule. Find the derivative of f(x) = 5x³.

  1. Identify the constant multiple: 5.
  2. Identify the function: x³.
  3. The rule says to pull the 5 out and just find the derivative of x³.
  4. f'(x) = 5 * d/dx [x³]
  5. Using the Power Rule, the derivative of x³ is 3x².
  6. f'(x) = 5 * (3x²) = 15x²

The Sum and Difference Rules

These are the rules that truly let us break apart big problems into small ones.

The Sum Rule: The derivative of a sum of functions is the sum of their derivatives. d/dx [f(x) + g(x)] = f'(x) + g'(x)

The Difference Rule: The derivative of a difference of functions is the difference of their derivatives. d/dx [f(x) - g(x)] = f'(x) - g'(x)

All this means is that you can take the derivative of a function term by term. If you see plus or minus signs, they act like walls, separating your function into distinct pieces. You can find the derivative of each piece on its own and then just connect them with the same plus or minus signs.

Putting It All Together: Differentiating Polynomials

This is the payoff. These four rules, combined with the Power Rule, unlock our ability to find the derivative of any polynomial in seconds.

Let's find the derivative of h(x) = 7x⁴ - 2x³ + x - 9.

We'll go term by term, just like assembling that gaming PC.

  1. 1
    First term
    7x⁴
    • This is a Constant Multiple (7) and a Power (x⁴).
    • d/dx [7x⁴] = 7 * d/dx [x⁴] = 7 * (4x³) = 28x³
  2. 2
    Second term
    -2x³
    • The minus sign comes with it. Think of it as + (-2x³).
    • d/dx [-2x³] = -2 * d/dx [x³] = -2 * (3x²) = -6x²
  3. 3
    Third term
    x
    • This is x¹. The constant multiple is 1.
    • d/dx [x] = 1 * x⁰ = 1 * 1 = 1. (Memorize this one: the derivative of x is 1.)
  4. 4
    Fourth term
    -9
    • This is a constant.
    • d/dx [-9] = 0

Now, the Sum/Difference Rule says we just put our pieces back together: h'(x) = 28x³ - 6x² + 1 - 0

So, h'(x) = 28x³ - 6x² + 1.

That's it. You took a big, scary-looking function and broke it down into four mini-problems that were easy to solve. This is the core of your day-to-day work with derivatives.

Worked examples

Let's walk through a few examples together. I'll show you my exact thought process, including how to set up the problem to make it easier.

Example 1

A Standard Polynomial

Problem: Find the derivative of f(x) = 4x³ - 7x² + 2x - 10.

Solution: This is a polynomial, so we can use our new rules to tackle it term by term. The plus and minus signs separate our terms.

  1. Differentiate the first term, 4x³:

    • This is a constant (4) times a function (x³). We use the Constant Multiple Rule and the Power Rule.
    • d/dx [4x³] = 4 * (3x³⁻¹) = 12x².

  2. Differentiate the second term, -7x²:

    • The constant is -7. The function is x².
    • d/dx [-7x²] = -7 * (2x²⁻¹) = -14x¹ = -14x.

  3. Differentiate the third term, +2x:

    • The constant is 2. The function is x (or x¹).
    • d/dx [2x] = 2 * (1x¹⁻¹) = 2 * (1x⁰) = 2 * 1 = 2.

  4. Differentiate the fourth term, -10:

    • This is a constant. The Constant Rule says its derivative is zero.
    • d/dx [-10] = 0.

  5. Combine the results:

    • Now we just assemble the derivatives of each term.
    • f'(x) = 12x² - 14x + 2 - 0

Final Answer: f'(x) = 12x² - 14x + 2

Key takeaway: Notice how we treated each term as its own small problem. The Sum and Difference rules give us permission to do that.

Example 2

Rewriting Before Differentiating

Problem: Find the derivative of g(t) = (5t⁴ - 2t) / 3.

Solution: This one looks a little different because of the division by 3. This is a classic spot to get stuck.

Where students get stuck: Many students see the fraction and either freeze or try to use a rule they haven't learned yet (the Quotient Rule). But we don't need it here!

The right move: The key is to rewrite the function first. Dividing the entire numerator by a constant is the same as dividing each term by that constant.

  1. 1
    Rewrite the function
    • g(t) = (5t⁴ / 3) - (2t / 3)
    • Let's make the constant multiples even clearer:
    • g(t) = (5/3)t⁴ - (2/3)t
  2. 2
    Differentiate term by term
    • Now it looks just like Example 1!
    • First term: d/dt [(5/3)t⁴] = (5/3) * (4t³) = (20/3)t³
    • Second term: d/dt [-(2/3)t] = -(2/3) * (1) = -2/3
  3. 3
    Combine the results
    • g'(t) = (20/3)t³ - 2/3

Final Answer: g'(t) = (20/3)t³ - 2/3

Key takeaway: A little bit of algebra before you start calculus can make your life much, much easier. Always ask yourself: "Can I rewrite this in a simpler form?"

Try it yourself

Time to get your hands dirty. Use the rules we've discussed to solve these. Don't just jump to the answer; think about which rule applies to each part of the function.

Problem 1: Find the derivative, dy/dx, for the function y = -x⁵ + 3x² - 11x + 4.

Hint: Go term by term. What is the derivative of -11x? What is the derivative of +4?

Problem 2: Find f'(x) for f(x) = 6√x + x/2.

Hint: You can't use the Power Rule on √x or x/2 directly. How can you rewrite both terms using exponents so the Power Rule applies? Think about fractional exponents and constant multiples.

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