Derivative Calculator

How to Use the Derivative Calculator

Enter any mathematical function using x as the variable. The calculator computes the symbolic derivative in real time as you type, applying standard differentiation rules automatically. You can use arithmetic operators (+, −, *, /, ^), built-in functions like sin(x), cos(x), tan(x), exp(x), ln(x), and sqrt(x), as well as the constants pi and e.

For example, entering x^3 + sin(x) produces 3 * x^2 + cos(x). The calculator handles nested functions through the chain rule, so sin(x^2) correctly differentiates to cos(x^2) * 2 * x.

Differentiation Rules Applied

The calculator implements these fundamental rules of differential calculus:

• Power rule: d/dx(xⁿ) = n · xⁿ⁻¹
• Product rule: d/dx(u · v) = u′ · v + u · v′
• Quotient rule: d/dx(u/v) = (u′ · v − u · v′) / v²
• Chain rule: d/dx(f(g(x))) = f′(g(x)) · g′(x)
• Trigonometric: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec² x
• Exponential: d/dx(e^x) = e^x, d/dx(a^x) = a^x · ln(a)
• Logarithmic: d/dx(ln x) = 1/x

Simplification

After computing the raw derivative, the calculator applies basic algebraic simplifications: eliminating multiplication by 1, addition of 0, constant folding, and removing zero terms. The result is a clean, readable expression. However, advanced simplifications like trigonometric identities or polynomial factoring are not performed, so some results may appear longer than hand-simplified forms.

Common Use Cases

Derivatives are central to calculus and appear throughout science, engineering, economics, and data analysis. Students use them to find rates of change, slopes of tangent lines, and to analyse function behaviour (increasing/decreasing intervals, concavity, and inflection points). In physics, the derivative of position is velocity and the derivative of velocity is acceleration. In machine learning, gradients (multivariable derivatives) drive optimisation algorithms. This calculator provides a quick way to verify hand-computed derivatives or explore how functions change.

Frequently Asked Questions

What is a derivative?

A derivative measures the instantaneous rate of change of a function. For f(x), the derivative f′(x) tells you the slope of the tangent line at each point on the curve.

Which differentiation rules does this calculator use?

It applies the power rule, product rule, quotient rule, chain rule, and standard formulas for trigonometric, exponential, and logarithmic functions.

Can this handle implicit differentiation?

No, this performs explicit symbolic differentiation with respect to x only. For implicit differentiation, apply the chain rule manually.

Why does the result look complicated?

Symbolic differentiation applies rules mechanically. Basic simplification is applied but advanced algebraic simplification is not performed.

What functions are supported?

You can use +, −, *, /, ^, parentheses, sin, cos, tan, exp, ln, log, sqrt, abs, and the constants pi and e.