d/dx Derivative Calculator

1. Enter the coefficients from highest degree to constant, separated by commas.
2. For example, enter 3,2,5,7 for 3x³ + 2x² + 5x + 7.
3. Click Calculate to get the derivative.
4. Review the step-by-step differentiation of each term.
5. Notice that the constant term drops out of the result.

What happens to the constant term?

The derivative of any constant is zero. This is because a constant does not change, so its rate of change is zero. When you differentiate 3x² + 5x + 7, the constant 7 produces 0 and simply disappears from the result. This is why you lose information when differentiating: you cannot recover the original constant from the derivative alone.

Can this calculator handle trigonometric or exponential functions?

No. This calculator uses the polynomial power rule, which applies only to terms of the form c·xⁿ. Trigonometric functions (d/dx sin x = cos x), exponentials (d/dx e^x = e^x) and logarithms (d/dx ln x = 1/x) require different rules that this tool does not cover.

What is the difference between derivative and integral?

Differentiation and integration are inverse operations. Differentiation breaks a function down into its rate of change, while integration builds up a function from its rate of change. If you differentiate a function and then integrate the result, you get back the original function plus a constant C. Together they form the Fundamental Theorem of Calculus.

What does the derivative tell you geometrically?

The derivative at a point gives the slope of the tangent line to the function at that point. Where the derivative is positive, the function is increasing; where it is negative, the function is decreasing; where it is zero, the function has a horizontal tangent (a local maximum, minimum or inflection point).

When differentiating polynomials, apply the power rule to each term: multiply the coefficient by the exponent, then reduce the exponent by 1. A common mistake is reducing the exponent without multiplying: d/dx [x³] = 3x², not x². Another common error is forgetting that constants vanish: many students leave the constant term in the derivative. To check your work, integrate the result: you should get back the original polynomial plus an arbitrary constant C. In optimization problems, set the derivative equal to zero to find critical points (where the function may have a maximum or minimum). The second derivative tells you whether a critical point is a maximum (second derivative negative) or a minimum (second derivative positive). In machine learning, the derivative of the loss function with respect to each parameter tells you how to adjust that parameter to reduce the loss, which is the core of gradient descent.