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Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
Enter a mathematical expression and click Calculate to see the symbolic derivative with step-by-step solution and interactive graph.
When a function depends on multiple variables, a partial derivative measures how the function changes with respect to one variable while holding all others constant. The notation uses the "curly d" to distinguish it from ordinary derivatives.
For :
To compute , treat as a constant and differentiate normally with respect to .
Collecting all partial derivatives into a vector gives the gradient:
The gradient points in the direction of steepest ascent. Its magnitude gives the rate of change in that direction. This is the foundation of gradient descent in machine learning.
Second-order partial derivatives include both "pure" and "mixed" partials:
Clairaut's theorem: If the mixed partials are continuous, then the order of differentiation doesn't matter: .
To compute partial derivatives with this calculator, change the variable field to the variable you want to differentiate with respect to. All other symbols are treated as constants automatically.
A partial derivative measures how a multivariable function changes with respect to one variable while holding all other variables constant. The notation uses the curly d symbol: means differentiate with respect to , treating , , etc. as constants.
Treat every variable except the one you are differentiating with respect to as a constant, then differentiate normally. For , the partial with respect to is ( treated as constant), and with respect to it is .
The gradient is the vector of all partial derivatives: . It points in the direction of steepest ascent and its magnitude is the rate of change in that direction. Gradient descent -- the core of machine learning optimization -- moves opposite to the gradient.
Clairaut's theorem states that if the mixed partial derivatives are continuous, the order of differentiation does not matter: . This symmetry simplifies many multivariable calculus computations.
Partial derivatives appear in optimization (finding critical points of multivariable functions), physics (heat equation, wave equation, Maxwell's equations), economics (marginal cost, marginal utility), and machine learning (backpropagation computes partial derivatives of the loss function).