# Derivative Calculator

Differentiate a function with mathjs and get the simplified derivative, its LaTeX code and optionally its value at a point. Choose the variable and an order from the first to the fifth derivative. Polynomials, powers, roots, exponentials, logarithms, trigonometric, inverse and hyperbolic functions are supported, and other letters are treated as constants. Helpful for calculus homework, checking work and preparing formulas for documents.

## Frequently asked questions

### How do I write the function?

Use standard math notation: x^2 for powers, * or a space for multiplication (2x and 2 x both work), / for division and parentheses for grouping. Functions take parentheses, as in sin(x), sqrt(x) or log(x). Note that xy is read as one variable named xy, so write x*y for a product.

### Which functions can be differentiated?

sin, cos, tan, sec, csc, cot and their inverses, the hyperbolic functions and their inverses, sqrt, cbrt, nthRoot, exp, log, log10, pow and abs. The constants pi, e, tau and phi are recognized. Any other function name is rejected with a message naming it.

### Is log the natural logarithm?

Yes, log(x) is the natural logarithm, so its derivative is 1 / x. Use log10(x) for the base 10 logarithm or log(x, b) for any base b. There is no separate ln function in this tool, so write log for the natural logarithm.

### What happens to other letters in the expression?

Every symbol other than the chosen variable is treated as a constant, so the derivative of a*x^3 with respect to x is 3 * a * x ^ 2. You can differentiate with respect to any variable name by changing the Variable field. Names that are built-in functions or constants, such as sin, pi or e, cannot be used as the variable.

### How do higher order derivatives work?

Set the order from 1 to 5 and the tool differentiates that many times, simplifying the expression after each step. The result row shows the notation used, for example d^2/dx^2 for the second derivative. Higher orders of complicated functions grow quickly, so the order is capped at 5.

### Can I evaluate the derivative at a point?

Yes, type a number or a constant expression such as 2, -0.5 or pi / 2 in the optional point field. The value is rounded to the decimal places you choose. If the derivative still contains other symbols such as a or k, the tool tells you which ones prevent a numeric value.

### What is the LaTeX row for?

It contains the derivative written as LaTeX code that you can paste into a document, a slide or a Markdown editor with math support. It describes the same simplified expression shown above it. The code is generated by mathjs toTex.

### Why does my result look different from my textbook?

Equivalent expressions can be written in many ways, for example 2 * cos(x) * sin(x) instead of sin(2x). mathjs applies its own simplification rules and does not apply trigonometric identities or factor polynomials. Both forms are mathematically equal.

### Are there limits on the input?

Expressions can be up to 2000 characters, the order up to 5, and each calculation has a 3 second time limit. Deeply nested functions can make the derivative grow very fast, and the tool stops with a clear message instead of running indefinitely. Assignments, function definitions and strings are not accepted.

### Does the tool handle the absolute value?

Yes, the derivative of abs(x) is returned as abs(x) / x, which equals the sign of x wherever x is not zero. The derivative does not exist at x = 0, and evaluating there reports an undefined result. The same caution applies to any point where the function is not differentiable.

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