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Derivatives

The difference quotient at the end of the Functions lesson measured the slope between two points a distance h apart. Let h shrink to nothing and it stops being a slope between two points and becomes the slope at one — that is the derivative, and everything here is either that definition or a shortcut for avoiding it. Learn the definition once so you know what the shortcuts are shortcuts for, then use the shortcuts.

Calculus — differentiation

Graphing calculator
What a derivative is
Three descriptions of the same number. Which one is useful depends on the question, so it is worth being able to move between them.
  • The definition

    The difference quotient with driven to nothing. "" means: what value does this head towards as gets small? Not what it equals at — there it is , which the Limits lesson before this one covers properly.

  • Slope of the tangent

    The straight line that just grazes the curve at that point. This is the picture to keep in your head.

  • Rate of change

    Position gives velocity, volume gives flow. Same number, read as a rate rather than a slope.

  • The notations

    All one thing. is an instruction — "differentiate what follows" — while is the answer. A derivative at a particular point is written or .

  • Where it fails to exist

    A derivative is a slope, and some points have no single slope. at has a corner: the slope is on the left and on the right, so there is no answer.

From the definition
Called differentiating from first principles. You will not do it often, but doing it a few times is what makes the rules that follow stop looking arbitrary. In every one of these the h on the bottom cancels — if it does not, the algebra above it went wrong.

The rules
Each one is provable from the definition, and none of them need re-deriving once you trust them. These are the whole toolkit for anything that is not a composite.
  • Constant

    A horizontal line has slope zero. Nothing about it changes.

  • Power rule

    Bring the exponent down in front, then knock one off it. Works for every real — negative, fractional, all of it — which is why rewriting a radical or a reciprocal as a power is usually the first move.

  • Constant multiple

    A constant factor sits out and waits. It never needs the product rule.

  • Sum and difference

    Differentiate term by term. This is the rule that makes polynomials easy.

  • Product rule

    "First times derivative of second, plus second times derivative of first" — in either order, since it is symmetric. It is emphatically not .

  • Quotient rule

    Order matters on top: the derivative of the top comes first. Swap them and every sign is wrong. Remember it as "low d-high minus high d-low, over low squared".

  • Exponential

    is its own derivative — the only function that is. Note the variable is in the exponent here, so the power rule has nothing to do with it.

  • Logarithm

    For another base, .

  • Trigonometric

    The minus belongs to cosine. Also , and these hold only when is in radians.

Two of these are about shape rather than about a particular function: the sum rule says differentiation splits over , and the product rule says it does not split over . Everything people get wrong here is one of those two being applied where the other belongs.

Which rule, and in what order
Most of the difficulty in a differentiation problem is decided before any differentiating happens. Read the structure first.

  1. Rewrite before you differentiate. Turn radicals into fractional powers and reciprocals into negative powers, and split a fraction whose bottom is a single term. is , which needs no quotient rule at all. This one habit removes most of the work.

  2. A sum of terms: go term by term. Differentiate each piece and keep the signs. Nothing about one term affects another.

  3. A constant times something: leave the constant alone. is . Using the product rule here is not wrong, just slower — the constant's derivative is zero and that term disappears.

  4. Two genuine functions multiplied: product rule. Genuine meaning both contain . For something small like , expanding first and using the power rule is often quicker; for there is no expanding to be done.

  5. One thing over another: quotient rule, or rewrite. If the denominator is a single power, a negative exponent beats the quotient rule every time. Keep the quotient rule for a genuine binomial below the line.

  6. Something inside something: chain rule. , , — anything wrapped in a shell needs the next lesson. Spotting that you are looking at a composite is the skill; the rule itself is one line.

Worked examples — the power rule
Most of these are really a rewriting exercise with one line of calculus at the end.

Worked examples — product and quotient
For these the structure genuinely needs the rule; there is no rewriting your way out.

Worked examples — using a derivative
A derivative is rarely the final answer. These are the two things a question usually wants next.

Where this goes wrong

  • Test it on something you know: has derivative , but . Differentiation splits over addition, never over multiplication.

  • Same test: has derivative 1, while . The shortcut is not a rule.

  • The power rule is for a variable base with a constant exponent. Here it is the other way round — constant base, variable exponent — and that is a different rule entirely. Check which one is moving before choosing.

  • The rule says subtract one, and subtracting one from gives . A negative exponent gets more negative, not less.

  • The minus is real and belongs to cosine. Sine differentiates cleanly to cosine; cosine differentiates to sine and picks up a sign.

  • Right shape, wrong order — and it gives the exact negative of the true answer every time. The derivative of the top always leads.

Practice
Rewrite first where it helps, then pick the rule.

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