Derivative of a Function Derivative of a Function The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = f(x), where f denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
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If y is a function of a type y = xn the formula for the derivative is: y = xn => y' = nxn-1 example: y = x3 y' = 3x3-1 = 3x2 y = x-3 y' = -3x-4 From the upper formula we can say for derivative y' of a function y = x = x1 that: if y = x then y'=1 y = f1(x) + f2(x) + f3(x) ...=> y' = f'1(x) + f'2(x) + f'3(x) ... This formula represents the derivative of a function that is sum of functions. example: If we have two functions f(x) = x2 + x + 1 and g(x) = x5 + 7 and y = f(x) + g(x) then y' = f'(x) + g'(x) => y' = (x2 + x + 1)' + (x5 + 7)' = 2x1 + 1 + 0 + 5x4 + 0 = 5x4 + 2x + 1 If a function is multiplication of two functions the derivate formula is: y = f(x).g(x) => y' = f'(x)g(x) + f(x)g'(x) If f(x) = C(C is a contstant), and y = f(x)g(x) y = Cg(x) y'=C'.g(x) + C.g'(x) = 0 + C.g'(x) = C.g'(x) y = Cf(x) => y' = C.f'(x)
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Increasing and Decreasing Functions Increasing and Decreasing Functions We all know that if something is increasing then it is going up and if it is decreasing it is going down. Another way of saying that a graph is going up is that its slope is positive. If the graph is going down, then the slope will be negative. Since slope and derivative are synonymous, we can relate increasing and decreasing with the derivative of a function. First a formal definition. A function is increasing on an interval if for any x1 and x2 in the interval then x1 < x2
implies
f(x1) < f(x2)
A function is decreasing on an interval if for any x1 and x2 in the interval then x1 < x2
implies
f(x1) > f(x2)
How does this relate to derivatives? Recall that the derivative is the limit f(x2) - f(x1) x2 - x1
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If x1 < x2, then the denominator will be positive. If also f(x1) < f(x2), then the numerator will be positive, hence the derivative will be positive. On the other hand if f(x1) > f(x2), then the numerator will be negative and the derivative will be negative. this leads us to the following theorem. Example :- Determine the values of x where the function f(x) = 2x3 + 3x2 - 12x + 7 Solution :- We first take the derivative f '(x) = 6x2 + 6x - 12 To determine where the derivative is positive and where it is negative, find the roots. Factor to get 6(x2 + x - 2) = 6(x - 1)(x + 2) Hence the change in sign can occur when x = 1 and x = -2 Now create some test values The derivative is positive outside of [-2,1] and is negative inside of [-2,1]. We can conclude that f is increasing outside of [-2,1] and decreasing inside of [-2,1]. The graph is shown below. We saw that the values of x such that the derivative is 0 was of special interest. Other points where there could be a change from increasing to decreasing is where the derivative is undefined. We call c a critical number if either f '(c) = 0 or f '(c) is undefined. Read More About Solve Differential Equation
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