How to Find a Derivative Without Losing the Plot

Calculus has a reputation it does not entirely deserve. Most of a first course is three rules applied over and over, plus enough patience to keep track of a minus sign.

What a derivative actually tells you

Suppose a car's distance from a start line is s(t) = 4t² metres after t seconds. Over the first 3 seconds it covers 4 × 9 = 36 metres, so its average speed was 12 m/s. It was not doing 12 m/s the whole time, though. The derivative s'(t) = 8t gives the speed at any instant: 8 m/s at one second, 24 m/s at three seconds.

That is the whole idea. An average tells you what happened across an interval. A derivative tells you what is happening at one point, which is what a speedometer shows.

The three rules that cover most problems

Power rule

Multiply by the exponent, then subtract one from it. So x⁵ becomes 5x⁴. Constants multiply straight through, and a constant sitting on its own disappears. Given f(x) = 7x⁵ - 2x² + 9, differentiate term by term: 35x⁴, then -4x, then nothing at all from the 9. So f'(x) = 35x⁴ - 4x.

Product rule

For two functions multiplied together, the derivative is the first times the derivative of the second, plus the second times the derivative of the first. Take f(x) = x² sin x. That gives x² cos x + 2x sin x. The common mistake is differentiating each piece separately and multiplying the results, which produces 2x cos x and is wrong.

Chain rule

For a function sitting inside another function, differentiate the outside and multiply by the derivative of the inside. With f(x) = (3x + 1)⁴, the outside gives 4(3x + 1)³ and the inside contributes 3, so f'(x) = 12(3x + 1)³. Forgetting that last multiplication is probably the single most common error in first-year calculus.

Two rules in one expression

Exam questions stack them. Take f(x) = x²(3x + 1)⁴. That is a product, and the second factor needs the chain rule inside it, so the derivative is x² × 12(3x + 1)³ added to (3x + 1)⁴ × 2x. Check it at x = 1, where the bracket is 4: the first term is 12 × 64 = 768, the second is 2 × 256 = 512, and the total is 1,280. Work outward one rule at a time and write each stage down. Mistakes come from holding two rules in your head at once, not from the rules being hard.

Using the derivative calculator

Type the expression with a caret for powers, so x^2 rather than a superscript, and put an asterisk between anything multiplied: 3*x*sin(x), not 3x sin x. Set the variable if the expression uses a letter other than x, and set the order if you want the second derivative rather than the first. Most versions will also evaluate the result at a value, which is how you get "the slope at x = 2 is 96" instead of just another formula.

Enter your own answer alongside and compare. If the two look different but graph identically, they are the same expression written differently, and a graphing calculator settles it in seconds.

Common questions

Why does the calculator's answer look nothing like mine? Usually because one of you expanded the brackets and the other did not. 12(3x + 1)³ and 324x³ + 324x² + 108x + 12 are the same function. Substitute a number into both and check. At x = 1: 12 × 64 = 768, and 324 + 324 + 108 + 12 = 768.

What is the derivative of a constant? Zero. A constant does not change, so its rate of change is nothing. That is why the 9 vanished above, and why an indefinite integral has to put a +C back on the end.

Is dy/dx different from f'(x)? Same thing, different notation. The Leibniz form dy/dx names the variable explicitly, which helps with the chain rule and with implicit differentiation. The f'(x) form is shorter. Physics often puts a dot over the letter when the derivative is with respect to time.

Where do quadratics come into this? Setting a derivative to zero finds turning points, and for a cubic that means solving a quadratic. If that step is rusty, the quadratic formula is worth a refresher before the calculus exam rather than during it.