# Derivative: is If f(x) is a realDerivative: is If f(x) is a real

Derivative: From our various experiment we realise that it is very important to know the values of certain parameters at various instance of time and how the values of these parameters change to find the rate of change. For example, the people responsible for maintaining a reservoir needs to know the time it will take to completely fill up by knowing the height of water at different instances of time. Thus, derivative or the rate of change is of great importance. Definition of derivative: Let f(x) is a real valued function and a be any point in the domain of Definition of f(x). The derivative of f(x) at a is defined by  provided that the limit exists. Derivative of f(x) at a is denoted by f?(a). So,  For example, to find the derivative of f(x) = 5x. at x = 10. We have   Or,  Or, . So, the derivative of f(x)=5x at x=10 is 5 Another example, to find out the derivative of f(x) = 3×2+2x-5 at x = 0 and also prove For this, we again have  Or,  Or, . The derivative of f(x) = 3×2+2x-5 at x = 0 is 2. Representing derivative graphically,  Let P (a, f(a)) and Q ((a+h), f(a+h)) be two neighbouring points on the graph of y = f(x). We know that   Now, triangle PQR shows that the ratio whose limit our concern here, is precisely equal to tan(QPR), the slope of PQ. As h tends to 0, the point Q tends to P and we have  Here the chord PQ tends to be a tangent to the curve. Therefore,  For a given function f(x) we can find the derivative at every point. If the derivative exists at every point, it also becomes a new function, the derivative of f(x). The formal definition of derivative is   If f(x) is a real valued function, then the function, which is the derivative of f(x) is defined as,  This is the definition of derivative aka first principle of derivative. It is clear from the above definition that, the domain of definition of f'(x) is as long as the limit exists. It is denoted as, , or if y = f(x), then . The derivative of f(x) at x = b is defined as, , Or if y = f(x), then . Example 1: Find the derivative of f(x) = x2. We know,   Or,  Example 2: Find the derivative of   We have,  Or,  Or, .  Algebra of derivative of functions:     Example 3: Find the derivative of f(x) = xn. The definition of the derivative function is   Or,  Now,  From the binomial theorem,  Or,  Therefore,  Or, . Since all the other term contains h. Derivative of polynomial: Let f(x) be a polynomial such that, , where ai’s are constant and . Therefore,  For example, the derivative of f(x) = 1+x2+x3+……..x50.at x = 1  Or, . Derivative of polynomial: Let f(x) = cos x  Or,  Or,  Or, Or,  Therefore,   For  f(x) = cos2 x  Or,  Or,  Or,  Or, .