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How to calculate the derivative

1. Enter the function f(x). For example x^3-2x or sin(x).
2. Enter the point x. The point at which the derivative is calculated.
3. Calculate. You get the value of f′(x) at the point, the second derivative and the tangent equation.

What does the derivative tell you?

The derivative describes the rate of change of the function, i.e. the slope of the curve at a given point. When the derivative is positive, the function increases; when it is negative, the function decreases; and when the derivative is zero, it may be a peak or a trough. The tangent is a line that touches the curve at the point and whose slope is exactly the value of the derivative.

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The calculator estimates the derivative numerically using the central difference, which gives a very accurate result for ordinary functions. The method works for all functions the browser supports, without you having to derive the derivative by hand. At very steep or discontinuous points the approximation may be off, so check the result for such functions.

Example: velocity and acceleration

In physics, the derivative describes the rate of change. If an object's position as a function of time is s(t) = 5t², its velocity is the derivative of position: s′(t) = 10t. At time t = 3 the velocity is 10 · 3 = 30. The derivative of velocity, in turn, is acceleration, here the constant 10. The same thinking works in economics (marginal cost is the derivative of the cost function) and in many other applications. The calculator gives the value of the derivative at the chosen point directly.

Derivointisäännöt lyhyesti

The basic building blocks are worth remembering: the derivative of a constant is 0, the derivative of the power xⁿ is n·xⁿ⁻¹, and the derivative of a sum is the sum of the derivatives. There are separate rules for the product and the quotient, and for a composite function the chain rule. The basic trigonometric derivatives are sin′ = cos and cos′ = −sin. Even though the calculator computes the derivative numerically, knowing the rules helps you check whether the result is of the right order of magnitude.

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One of the most important applications of the derivative is finding extrema. At a function's peaks and troughs the tangent is horizontal, i.e. the derivative is zero. By solving f′(x) = 0 you find the possible extremum points, and the change in the sign of the derivative tells whether it is a maximum or a minimum. If the derivative changes from positive to negative, the function has a peak; the opposite gives a trough.

This is used in optimisation everywhere: finding the lowest price, the greatest yield or the smallest material consumption. The second derivative completes the picture by telling about the curvature of the curve — a positive second derivative means a convex (cup-shaped) form opening upward. When you calculate the value of the derivative at a point with this calculator, you get the slope, and by examining it at different points you get a sense of where the function increases, decreases and reaches its extrema.

Good to know about the derivative

• The derivative of a constant function is zero, because its value does not change.
• The zeros of the derivative are possible extremum points (peaks or troughs).
• A positive derivative means increase, a negative one decrease.
• The slope of the tangent at a point is exactly the value of the derivative at that point.

संक्षेप में

The derivative calculator gives a function's derivative at a point, the second derivative and the tangent equation. Enter the function and a point, and you get the slope and the tangent without differentiating by hand. The derivative tells the function's rate of change and helps find extrema, which is useful both in study and in optimisation. Angles are interpreted in radians, and the tool computes the value numerically, so it works with all functions the browser supports without derivation by hand.

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