Derivative: Differentiate a function—Wolfram Documentation (2024)

FAQs

How do you differentiate a function from a derivative? ›

To differentiate y=f(g(x)) y = f ( g ( x ) ) , let u=g(x) u = g ( x ) so that we have y as a function of u , y=f(u) y = f ( u ) . Then the chain rule says: dydx=dydu×dudx d y d x = d y d u × d u d x Once you have worked this out, you replace u by g(x) and your answer is now in terms of x .

You can define the derivative in Mathematica of a function f of one argument simply by an assignment like f'[x_]=fp[x]. This defines the derivative of f(x) to be fp(x). In ... Derivatives in Mathematica work essentially the same as in standard mathematics.

How Wolfram|Alpha calculates derivatives. Wolfram|Alpha calls Wolfram Languages'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 and so on.

The Wolfram Language's differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user.

In the Wolfram System, D[f,x] gives a partial derivative, with all other variables assumed independent of x. Dt[f,x] gives a total derivative, in which all variables are assumed to depend on x. In both cases, you can add an argument to give more information on dependencies.

You can check certain values, like the saddle points, extremal points and local minima/maxima by setting the first derivative equal to zero/deriving further and checking these derivatives too. If you found them right, putting the values into the original function plus/minus some Δx should make things clear.

A derivative is a function that you can use to calculate the slope of another function at any given point. If you have a function like f(x)=2x f ( x ) = 2 x , the slope is 2 everywhere, so the derivative is just f′(x)=2 f ′ ( x ) = 2 .

The first notation is to write f′(x) for the derivative of the function f(x). This functional notation was introduced by Lagrange, based on Isaac Newton's ideas. The dash in f′(x) denotes that f′(x) is derived from f(x). The other notation is to write dydx.

How to Prove a Function is Differentiable? A function can be proved differentiable if its left-hand limit is equal to the right-hand limit and the derivative exists at each interior point of the domain.

What is the function of the Wolfram? ›

The functions you define in the Wolfram Language are essentially procedures that execute the commands you give. You can have several steps in your procedures, separated by semicolons. The result you get from the whole function is simply the last expression in the procedure.

The top graph is the original function, f(x), and the bottom graph is the derivative, f'(x). What do you notice about each pair? If the slope of f(x) is negative, then the graph of f'(x) will be below the x-axis. If the slope of f(x) is positive, then the graph of f'(x) will be above the x-axis.

The derivative of a function f(x) is the function whose value at x is f′(x). The graph of a derivative of a function f(x) is related to the graph of f(x). Where f(x) has a tangent line with positive slope, f′(x)>0. Where f(x) has a tangent line with negative slope, f′(x)<0.

Find where one is always increasing, and the other always corresponds to be positive, and find where the function is decreasing, to find the corresponding negative function. The graph that's positive and negative while the other is increasing and decreasing is a derivative function.

While differential and derivative are related, they are not the same thing. The main difference between differential and derivative is that a differential is an infinitesimal change in a variable, while a derivative is a measure of how much the function changes with respect to its input.