Derivative Calculator
Differentiate any function of x symbolically. Get f′(x) and f″(x) in simplified form, their value at a point, and a numerical check of the result.
Derivatives
- f(x)
- x^3 - 5*x^2 + 2*x - 8
- First derivative f′(x)
- 3*x^2 - 10*x + 2
- Second derivative f″(x)
- 6*x - 10
Numerical cross-check
The symbolic derivative is checked against a central-difference estimate (f(x+h) − f(x−h)) / 2h.
- symbolic f′(2) = -6
- central difference = -6.0000000035
- difference = 3.5030000000e-9
The two agree, so the symbolic result is behaving as expected here.
What is the Derivative Calculator?
The ByteTools Derivative Calculator parses the function you type into an expression tree and differentiates it symbolically, applying the power, sum, product, quotient and chain rules along with the standard derivatives of sin, cos, tan, ln, exp and sqrt.
- True symbolic differentiation, not a numerical approximation
- Power, product, quotient and chain rules plus standard derivatives
- Second derivative computed and simplified automatically
- Values of f, f′ and f″ at any point you choose
- Every result cross-checked against a central-difference estimate
- Runs entirely in your browser — nothing is uploaded
How to use the Derivative Calculator
- 1
Type your function in the f(x) box using x, numbers and functions such as sin, cos, ln, exp or sqrt.
- 2
Read the simplified first and second derivatives in the Derivatives panel.
- 3
Enter a value in the 'Evaluate the derivative at x' box to get f(a), f′(a) and f″(a).
- 4
Check the numerical cross-check panel to confirm the symbolic slope matches the central-difference estimate.
- 5
Click Copy result to take the derivatives and values with you.
About the Derivative Calculator
The ByteTools Derivative Calculator parses the function you type into an expression tree and differentiates it symbolically, applying the power, sum, product, quotient and chain rules along with the standard derivatives of sin, cos, tan, ln, exp and sqrt. It then simplifies what comes back, so you read 3x² − 10x + 2 rather than a tangle of unfolded products.
You also get the second derivative, and the value of f, f′ and f″ at any x you choose. Every symbolic answer is cross-checked against a central-difference estimate of the same slope, so if the two disagree the page tells you rather than quietly presenting a number that looks fine.
Everything runs in your browser with JavaScript. The function you type is never uploaded, nothing is stored, and the whole page keeps working offline once loaded.
Frequently asked questions
How do you find the derivative of a function?
You apply differentiation rules to each part of the expression: the power rule for xⁿ, the product and quotient rules for things multiplied or divided, and the chain rule for functions inside functions. This calculator applies all of them symbolically and simplifies the result for you.
What is the difference between the first and second derivative?
The first derivative f′(x) gives the slope of the curve, so it tells you where the function rises and falls. The second derivative f″(x) is the derivative of the slope, which describes concavity and helps identify maximums, minimums and inflection points.
Can this calculator show the steps of differentiation?
It shows the simplified symbolic result for f′(x) and f″(x) rather than a written-out rule-by-rule walkthrough. The numerical cross-check panel does show the working behind the verification, comparing the symbolic slope with a central-difference estimate at your chosen point.
Why does the derivative say it is not defined at my chosen x?
Some functions have points where no slope exists — ln(x) at x ≤ 0, sqrt(x) at negative x, or 1/x at zero. The calculator reports these honestly with a dash instead of printing a meaningless number.
What functions can I type into this derivative calculator?
You can use x, whole and decimal numbers, the operators + − * / ^, brackets, and the functions sin, cos, tan, asin, acos, atan, ln, log, exp, sqrt and abs. The constants pi and e are also recognised, and 2x is read as 2 times x.
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