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How to Find Derivatives
The rules you actually need, one worked example for each, and how to use derivatives to find highs and lows.
The derivative of a function tells you how fast it is changing. On a graph, the derivative at a point is the slope of the tangent line there. If f(x) is the position of a car, f′(x) is its speed. You can find almost any derivative with five rules.
1. The power rule
To differentiate a power of x, bring the exponent down in front and subtract one from it:
Constants disappear (their derivative is 0), constant multipliers stay, and you differentiate a sum one term at a time.
Example: Differentiate 3x⁴ − 2x² + 7.
3·4x³ − 2·2x + 0, so the derivative is 12x³ − 4x.
Try it in CasCalc →The power rule also works for roots and fractions once you write them as powers: √x = x1/2 has derivative 1/(2√x), and 1/x = x−1 has derivative −1/x².
2. The product rule
When two functions are multiplied, the derivative is first times derivative of second, plus second times derivative of first:
Example: Differentiate x² sin(x).
u = x² and v = sin(x), so u′ = 2x and v′ = cos(x). The derivative is 2x·sin(x) + x²·cos(x).
Try it in CasCalc →3. The quotient rule
For a fraction, use low d-high minus high d-low, over low squared:
Example: Differentiate x / (x² + 1).
u = x, u′ = 1, v = x² + 1, v′ = 2x. That gives ((x² + 1) − 2x²) / (x² + 1)² = (1 − x²) / (x² + 1)².
Try it in CasCalc →4. The chain rule
For a function inside another function, differentiate the outside (leaving the inside alone), then multiply by the derivative of the inside:
Example: Differentiate sin(3x²).
The outside is sin, whose derivative is cos. The inside is 3x², whose derivative is 6x. So the answer is 6x·cos(3x²).
Try it in CasCalc →The chain rule is the one students forget most often. Whenever you see something more complicated than plain x inside a function, ask yourself: what is the derivative of the inside?
5. Common derivatives to memorize
| Function | Derivative |
|---|---|
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| tan(x) | sec²(x) |
| ex | ex |
| ax | ax·ln(a) |
| ln(x) | 1/x |
Combined with the chain rule, these cover most problems: the derivative of e2x is 2e2x, and the derivative of ln(x² + 1) is 2x/(x² + 1).
Try ln(x² + 1) in CasCalc →Using derivatives: finding maximums and minimums
At the top of a hill or the bottom of a valley, the tangent line is flat, so the derivative is zero. To find these turning points, set the derivative equal to zero and solve.
Example: Find the turning points of y = x³ − 3x.
The derivative is 3x² − 3. Setting 3x² − 3 = 0 gives x² = 1, so x = −1 or x = 1.
At x = −1, y = −1 + 3 = 2. At x = 1, y = 1 − 3 = −2. The graph has a local maximum at (−1, 2) and a local minimum at (1, −2).
Graph it and see the turning points →CasCalc marks maximums and minimums on the graph automatically, which is a quick way to check your algebra.
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