![]() = sin (x 2) + C Antiderivative Product Rule Substitute this into the integral, we have Let us see an example and solve an integral using this antiderivative rule. ![]() The antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x)) dx. The chain rule of derivatives gives us the antiderivative chain rule which is also known as the u-substitution method of antidifferentiation. We know that antidifferentiation is the reverse process of differentiation, therefore the rules of derivatives lead to some antiderivative rules. Please do not confuse this power antiderivative rule ∫x n dx = x n+1/(n + 1) + C, where n ≠ -1 with the power rule of derivatives which is d(x n)/dx = nx n-1. Using the antiderivative power rule, we can conclude that for n = 0, we have ∫x 0 dx = ∫1 dx = ∫dx = x 0+1/(0+1) + C = x + C. Let us consider some of the examples of this antiderivative rule to understand this rule better. ![]() This rule is commonly known as the antiderivative power rule. Now, the antiderivative rule of power of x is given by ∫x n dx = x n+1/(n + 1) + C, where n ≠ -1. The antiderivative rules are common for types of functions such as trigonometric, exponential, logarithmic, and algebraic functions. We will discuss the rules for the antidifferentiation of algebraic functions with power, and various combinations of functions. For higher-order derivatives, certain rules, like the general Leibniz product rule, can speed up calculations.In this section, we will explore the formulas for the different antiderivative rules discussed above in detail. Additionally, D uses lesser-known rules to calculate the derivative of a wide array of special functions. It uses well-known rules such as the linearity of the derivative, product rule, power rule, chain rule and so on. 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. For example, it is used to find local/global extrema, find inflection points, solve optimization problems and describe the motion of objects. The derivative is a powerful tool with many applications. Īs an example, if, then and then we can compute. Geometrically speaking, is the slope of the tangent line of at. ![]() This limit is not guaranteed to exist, but if it does, is said to be differentiable at. Note for second-order derivatives, the notation is often used.Īt a point, the derivative is defined to be. These are called higher-order derivatives. When a derivative is taken times, the notation or is used. Given a function, there are many ways to denote the derivative of with respect to. What are derivatives? The derivative is an important tool in calculus that represents an infinitesimal change in a function with respect to one of its variables.
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