Advanced Math

Derivative Calculator

Enter a function of x. Every rule used is shown — constant, power, sum, product, quotient and chain rules, plus the standard derivatives of sin, cos, tan, e^x, ln, log (base 10) and sqrt — and the result is simplified. Domain restrictions such as ln(x) needing x > 0 are listed.

Inputsd/dx f(x)

f(x)Examples: 3x^4 − 2x + 7 · x*sin(x) · sin(x)/x · e^(2x) · ln(x^2 + 1) · sqrt(x). Use * or just write 3x. log means base 10.

Step-by-step

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Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

7 tools in Advanced Math.Back to the tool directory →

Worked example

Differentiate f(x) = x² sin(x).

1.
Start with the function.
```
f(x) = x^2·sin(x)
```
2.
Work from the innermost parts outward, applying one rule at a time.
3.
Power rule: d/dx[u^n] = n·u^(n−1). Bring down the exponent 2 and reduce it by 1.
```
d/dx[x^2] = 2x
```
4.
d/dx[sin(u)] = cos(u).
```
d/dx[sin(x)] = cos(x)
```
5.
Product rule: (uv)′ = u′v + uv′ with u = x^2 (u′ = 2x) and v = sin(x) (v′ = cos(x)).
```
d/dx[x^2·sin(x)] = 2x·sin(x) + x^2·cos(x)
```
6.
Simplify by collecting numbers and like terms.
```
f′(x) = 2x·sin(x) + x^2·cos(x)
```

Answer: f′(x) = 2x·sin(x) + x^2·cos(x)

Note: Handles powers, sin, cos, tan, eˣ, ln, log and sqrt with the product, quotient and chain rules. Not xˣ.

Use “Load worked example” above to run this in the solver.

Related formulas

The formulas this tool uses, each explained with symbols, conditions and its own example in the Formula Library.

Power rule (derivatives)d/dx (xⁿ) = n·xⁿ⁻¹ Product ruled/dx (u·v) = u′v + uv′Quotient ruled/dx (u ÷ v) = (u′v − uv′) ÷ v² Chain ruled/dx f(g(x)) = f′(g(x)) · g′(x)Standard derivatives(sin x)′ = cos x, (cos x)′ = −sin x, (eˣ)′ = eˣ, (ln x)′ = 1/x