Examples · Calculus
Derivative Examples: 10 Worked Problems
Ten derivatives, ordered so each one adds exactly one new rule to the last.
Power rule, including the awkward exponents
The first three derivative examples need nothing but the power rule and a rewrite.
- 15x⁴ − 4power rule on each term; the 7 differentiates to 0
- x−²rewrite with a negative exponent
- −2x−³power rule, which is now straightforward
- x^½a square root is a fractional power
- ½x^−½ = 1 ÷ (2√x)power rule, then tidy
Chain rule
Differentiate the outside, then multiply by the derivative of the inside.
- cos(2x) · 2outside gives cos; inside 2x gives 2
- 2cos(2x)tidy up
- 5(2x + 1)⁴ · 2power rule outside, derivative of the inside is 2
- 10(2x + 1)⁴tidy up
- e^(x²) · 2xe^u keeps itself, times the inside derivative
- 2x e^(x²)tidy up
Product and quotient rules
- u′v + uv′product rule
- 2x sin x + x² cos xsubstitute the four pieces
- 1 · ln x + x · (1/x)product rule
- ln x + 1x · (1/x) is 1
- [(x − 1) − (x + 1)] ÷ (x − 1)²quotient rule
- −2 ÷ (x − 1)²the numerator collapses
Example 8 is worth remembering: the derivative of x ln x is ln x + 1, and it appears constantly in integration by parts.
Two rules at once
The last one needs the product rule on the outside and the chain rule within it.
- 2x sin(3x) + x² · [sin(3x)]′product rule first
- [sin(3x)]′ = 3cos(3x)chain rule on the second factor
- 2x sin(3x) + 3x² cos(3x)put it together
The order matters. The outermost operation is the product, so that rule comes first, and the chain rule appears where it is needed inside.
The three standard functions in one place
The worked examples above lean on three families, and it is worth seeing a trig derivative, an exponential and a logarithm side by side with the same inside function.
| Function | Derivative | Which rule |
|---|---|---|
| sin(3x) | 3cos(3x) | chain |
| e^(3x) | 3e^(3x) | chain |
| ln(3x) | 1/x | chain, then it simplifies |
The last row is the surprising one. The chain rule gives 3 · 1/(3x), and the threes cancel — so the derivative of ln(3x) is the same as the derivative of ln x. Any constant multiplier inside a logarithm disappears when you differentiate, because it is an additive constant once the log is expanded.
That is also a reminder to simplify a composite function's derivative rather than leaving it in raw form: the tidy version says something the raw one hides.
Checking a derivative without re-doing it
Pick a value of x and compare the derivative with a difference quotient: work out (f(x + h) − f(x − h)) / 2h for a small h, say 0.001.
| Example | At x = 1 | Derivative | Difference quotient |
|---|---|---|---|
| sin(2x) | 2cos 2 | −0.832 | −0.832 ✓ |
| (2x+1)^5 | 10·3⁴ | 810 | 810.0 ✓ |
| x ln x | ln 1 + 1 | 1 | 1.000 ✓ |
This is the check the calculator on this site uses on itself, across every expression it supports. It works because a numeric estimate knows nothing about the rules, so it cannot repeat the same mistake.
Questions about derivative examples
What is the derivative of √x?
1/(2√x). Rewrite the root as x^½, apply the power rule to get ½x^−½, then tidy.
Why does (2x + 1)⁵ get a factor of 2?
The chain rule. After the power rule gives 5(2x + 1)⁴, you multiply by the derivative of the inside, which is 2.
What is the derivative of x ln x?
ln x + 1, by the product rule. The second term is x · (1/x), which simplifies to 1.
How can I check a derivative?
Compare it numerically. Evaluate (f(x + h) − f(x − h))/2h at some x with h small; it should be close to your derivative at that point.