Wolfram|Alpha is a great calculator for first, second, and third derivatives; derivatives at a point; and partial derivatives. Learn what derivatives are and how Wolfram|Alpha calculates them.
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Given a function `f(x)`, there are many ways to denote the derivative of `f` with respect to `x`. The most common ways are `(df)/dx` and `f'(x)`. When a derivative is taken `n` times, the notation `(d^n f)/(dx^n)` or `f^n(x)` is used. These are called higher order derivatives. Note for second order derivatives, the notation `f''(x)` is often used.
At a point `x = a`, the derivative is defined to be `f'(a) = lim_(h->0)(f(a+h)-f(a))/(h)`. This limit is not guaranteed to exist, but if it does, `f(x)` is said to be differentiable at `x = a`. Geometrically speaking, `f'(a)` is the slope of the tangent line of `f(x)` at `x = a`.
As an example, if `f(x) = x^3`, then
`f'(x) = lim_(h->0)((x + h)^3 - x^3)/(h) = 3 x^2`,
and then we can compute `f''(x)`:
`f''(x) = lim_(h->0)(3(x + h)^2 - 3x^2)/(h) = 6 x`.
The derivative is a powerful tool with many applications. For example, it is used to find local/global extrema, find inflection points, solve optimization problems, and describe the motion of objects.
Wolfram|Alpha calls Mathematica's `D` function, which uses a table of identities much larger than one would find in a standard calculus textbook. It uses ”well known” rules such as the linearity of the derivative, product rule, power rule, chain rule, so on. Additionally, `D` uses ”lesser known” rules to calculate the derivative of a wide array of special functions. For higher order derivatives, certain rules like the general Leibniz product rule can speed up calculations.