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Forgot Password Please enter the email address and we’ll send you an email containing instructions for changing your password. To create your new password, just click the link in the email we sent you. The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point. In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.
The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. Differentiation has applications to nearly all quantitative disciplines. The reaction rate of a chemical reaction is a derivative. Derivatives are frequently used to find the maxima and minima of a function.
Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. A general function is not a line, so it does not have a slope. A closely related notion is the differential of a function. Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Since the 17th century many mathematicians have contributed to the theory of differentiation.
This is called the second derivative test. Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization. This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points. In higher dimensions, a critical point of a scalar valued function is a point at which the gradient is zero. One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line.
But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear. Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. The mean value theorem gives a relationship between values of the derivative and values of the original function. In practice, what the mean value theorem does is control a function in terms of its derivative. This means that its tangent line is horizontal at every point, so the function should also be horizontal.
All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function. The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. Taylor’s theorem gives a precise bound on how good the approximation is.
The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions. Some natural geometric shapes, such as circles, cannot be drawn as the graph of a function.