Derivative Calculator

Type a function of x and the calculator finds its derivative symbolically using the power, product, quotient and chain rules, then simplifies the result. It shows the derivative of each term, the second derivative, the slope at any point and the equation of the tangent line there, and plots the function, its derivative and the tangent together. Every answer is checked numerically.

What a derivative tells you

The derivative f′(x) measures how fast a function changes: the slope of its graph at each point. Where f′(x) is positive the function is increasing, where it is negative the function is decreasing, and where it is zero the graph has a flat tangent — often a maximum or minimum. In physics the derivative of position is velocity; in economics the derivative of cost is marginal cost. The second derivative f″(x) measures how the slope itself changes, telling you about curvature (concave up or down).

Differentiation rules

How to use the calculator

1. Type the function using x as the variable, for example x^3 − 4x + 2sin(x).
2. Read the derivative, the second derivative and the table showing each term, the rule used and its derivative.
3. Enter a value of x to get the slope at that point and the tangent line equation.
4. Set the graph range to see f(x) (solid), f′(x) (dotted) and the tangent (dashed red).
5. Check the verification line — every derivative is compared with a numerical estimate.

Worked example

For f(x) = x³ − 4x + 2 sin x, differentiate term by term: the power rule gives 3x² for x³, the constant multiple rule gives −4 for −4x, and 2 sin x becomes 2 cos x. So f′(x) = 3x² − 4 + 2 cos x and f″(x) = 6x − 2 sin x. At x = 1: f(1) = 1 − 4 + 2 sin 1 ≈ −1.3171, and f′(1) = 3 − 4 + 2 cos 1 ≈ 0.0806, so the tangent line is almost flat: y ≈ 0.0806x − 1.3977.

The chain rule in practice

The chain rule handles a function inside another function: differentiate the outside, keeping the inside unchanged, then multiply by the derivative of the inside. For sin(x²), the outside is sin and the inside is x², so the derivative is cos(x²)·2x. For (3x + 1)⁵ it is 5(3x + 1)⁴·3 = 15(3x + 1)⁴. The calculator applies the chain rule automatically wherever a function contains more than plain x.

Finding maxima and minima

1. Find f′(x) and set it equal to zero.
2. Solve for x — these are the critical points.
3. Use f″(x): positive means a local minimum, negative a local maximum.
4. Check the endpoints if the domain is limited.
5. Use the graph to confirm: the dotted derivative crosses zero exactly where the function turns.

Derivatives in the real world

Common mistakes

• Forgetting the chain rule: the derivative of sin(3x) is 3cos(3x), not cos(3x).
• Treating the product rule as “multiply the derivatives”: (fg)′ is f′g + fg′, not f′g′.
• Mixing up the order in the quotient rule — it is f′g − fg′ on top.
• Differentiating constants as if they were variables: the derivative of π² or e³ is 0.
• Dropping negative signs from the derivative of cos x.

Higher derivatives

Differentiating again gives the second derivative f″(x), and then the third, and so on. The second derivative describes concavity: where f″(x) > 0 the graph curves upward like a cup, and where f″(x) < 0 it curves downward like a cap. A point where f″ changes sign is an inflection point. In physics, the second derivative of position is acceleration. The calculator shows f″(x) automatically; paste it back in to find the third derivative.

About the simplified form

Computer algebra can write the same derivative in many equivalent ways. This calculator applies common simplifications — combining constants, removing zeros and ones and cancelling simple factors — but it does not expand or factor everything, so your textbook answer may look different while being equal. To compare, evaluate both at a couple of values of x: equal values mean equal derivatives.

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