1st and 2nd derivative of sin(x)
- Input
- d/dx sin(x), d²/dx² sin(x)
- Expected output
- cos(x), -sin(x)
The second derivative of sine is sine itself with the sign flipped.
Carregando ferramenta… / Loading tool…
derivative of sine and cosine
The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These are two of the most used derivatives in calculus, and this page opens with sin(x) already selected, showing the rule's single step and the two curves plotted together.
The second derivative of sine is sine itself with the sign flipped.
After 4 successive differentiations, cosine returns to itself, closing the cycle.
Polynomials, trigonometric functions (sine, cosine, tangent and their inverses), hyperbolic functions, exponentials and logarithms, including combinations built from sums, subtraction, products, quotients and composition (the chain rule). Factorial, the remainder operator (%) and min/max are not elementarily differentiable, and the calculator says so instead of risking a wrong result.
It is a direct consequence of the two functions' geometric definition on the unit circle: near x = 0, sin(x) is rising (positive slope, equal to cos(0) = 1), while cos(x) is coming down from its maximum value (negative slope, equal to -sin(0) = 0, turning negative right after).
Yes, type whatever combination you want in the function field (it accepts any valid expression), and the sum or product rule automatically kicks in, combining the separately calculated derivatives of sin(x) and cos(x).
Examples by rule
cos(x)
\cos\left(x\right)
No formula renderer on this page: copy the LaTeX above and paste it into an editor that displays it, like a document or a note.
Derivative of sine (chain rule): the derivative of sin(u) is cos(u) * u'.
Derivative of the variable with respect to itself: the result is 1.
Everything runs in your browser: the expression you type, the symbolic differentiation and the graph never go through any server.