Derivative Calculator

Results

Read as: $f(x)=x^2\sin \left(x\right)$

How to Use the Derivative Calculator

1. 1. Type the function. Write it the way you would on paper: 3x^4 - 2x^2 + 7x, x^2 sin(x), (x^2+1)/(x-1), e^(-x^2). Multiplication can be left out between a number or letter and what follows, so 2x and x sin x work.
2. 2. Choose the variable and the order. The derivative is taken with respect to x unless you pick another letter; every other letter is treated as a constant. Order 2 gives the second derivative, and so on up to 5.
3. 3. Add a point if you want a number. Enter 2, -1.5 or pi/4 to get the function and its derivative evaluated there, exactly where the algebra allows (1/2, 2sin(1) + cos(1)) and as a decimal.
4. 4. Read the working. The answer appears first, then every derivative up to the order you chose, then one line per subexpression naming the rule used and the derivative it produced.

The Rules the Calculator Applies

Worked Examples

Product and chain rules together. For f(x) = x² sin(x), the product rule with u = x² and v = sin x gives u′v + uv′ = 2x sin x + x² cos x. At x = 1 that is 2 sin 1 + cos 1 ≈ 2.2232443. For f(x) = sin(x²) the outer function is sine and the inside is u = x², so f′(x) = cos(x²)·2x = 2x cos(x²).

Quotient rule, then higher orders. For f(x) = (x² + 1)/(x − 1), u′v − uv′ = 2x(x − 1) − (x² + 1) = x² − 2x − 1, so f′(x) = (x² − 2x − 1)/(x − 1)². At x = 3 that is 2/4 = 1/2. Differentiating again, the numerator of the second derivative has a factor of (x − 1) that cancels, leaving f″(x) = 4/(x − 1)³ — the calculator cancels such factors so higher derivatives come out in their short form.

How Your Input Is Read

• Multiplication. 2x, 3(x+1), (x+1)(x-1) and x sin x multiply. A bare number cannot follow a letter or bracket (x2 is refused, not guessed): write x*2 or 2x.
• Division. Implicit multiplication ranks with * and /, left to right, so 1/2x means (1/2)·x. Write 1/(2x) for the other reading. The “Read as” line shows exactly what was understood.
• Powers. -x^2 is −(x²), 2^-x is 2 raised to −x, and a^b^c is a^(b^c). sin^2(x) means (sin x)². sin^-1(x) is refused as ambiguous — type asin(x) or 1/sin(x).
• Functions without brackets. sin x, ln x^2 (that is ln of x²) and sin 2x take one simple argument; sin x (x+1) is sin(x) times (x+1).
• Names. Variables are single letters, e is Euler's number, pi or π is π, exp(x) is eˣ, ln is the natural logarithm and log is base 10. log(x, 2) is the base-2 logarithm. Angles are in radians.

What to Keep in Mind

Results are simplified to a conventional form — like terms combined, polynomials expanded, shared factors pulled out, and common factors cancelled between numerator and denominator — but a different-looking answer can still be correct. Check by evaluating both at a point. Cancelling a factor such as (x − 1) can quietly widen the domain at x = 1, as in any algebra system, so a derivative is only claimed where the original function is defined.

The derivative of |u| is u·u′/|u|, which does not exist where u = 0; the graph shows the corner. The tool differentiates explicit functions of one chosen variable, treating other letters as constants (which is how a partial derivative is taken); it does not do implicit differentiation.

Frequently Asked Questions

How do you find a derivative step by step?

Identify the outermost structure of the expression and apply the matching rule: split sums term by term, pull out constant factors, use the product or quotient rule when two parts both contain the variable, and the chain rule when a function has something other than x inside it. Then repeat on each piece. The working above follows exactly that order, outermost first.

When do I use the chain rule, and when the product rule?

Use the product rule when two factors multiplied together both depend on x, as in x² sin x. Use the chain rule when one function sits inside another, as in sin(x²) or e^(3x). Many problems need both, like x² sin(3x).

What is the second derivative used for?

It is the derivative of the derivative, so it measures how fast the slope itself changes. A positive second derivative means the graph is concave up (curving like a cup); a negative one means concave down. At a point where f′ = 0, the sign of f″ tells a maximum from a minimum.

Can I take a partial derivative?

Yes, in the sense that matters for computing one. Type the function with several letters, such as x^2 y + sin(y x), and choose the variable to differentiate with respect to; the other letters are held constant. The graph is hidden for functions of more than one variable.

Why is the derivative of |x| undefined at 0?

The graph of |x| has a sharp corner at the origin: the slope is −1 just to the left and +1 just to the right, so there is no single tangent line. Away from 0 the derivative is x/|x|, which equals −1 for negative x and +1 for positive x.

Is log the natural logarithm or base 10?

Here log is base 10, as on a scientific calculator, and ln is the natural logarithm; subjects differ on this, so check your textbook. For any other base type log(x, b). The derivative of log(x) is 1/(x ln 10), and of ln(x) is 1/x.

Why does my answer look different from the textbook's?

Equivalent expressions can be written many ways: (x + 1)eˣ and xeˣ + eˣ are the same function. If the form differs, evaluate both at a point or two — the values at your chosen point let you do that — rather than assuming one is wrong.