Nth Derivative Of a^x | Engg. Mathmatics

    

                    Nth Derivative Of a^x

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In this tutorial you will learn how to find  Nth Derivative Of a^x by successive differentiatin method. in successive differentiation method we will keep Differentiating 1/(ax+b) until we get desiered nth Derivatives of a^x. Complete solution is given below.

                            Let         Y = a^x
                                          Y1 = a^x log a

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Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives.

If we’re given a function y of a real variable and a point a in its domain, the value of f’(a) is the slope of the unique—if it exists— If this function is differentiable at each point of a domain interval, we can take its derivative too, giving you another function y=f”(x). Given the original function is nice enough, we can carry this process an indefinite number of times, successively differentiating the original function. The successive differentiation process eventually gives us y=0 (constant function). On the other hand, if f(x)=sin(x), we have f’(x)=cos(x), f’’(x)=-sin(x), f’’’(x)=-cos(x), and f’’’’(x)=sin(x). Successive differentiation cycles around these four functions, since f’’’’(x)=f(x). In another example, all successive differentiations of the function f(x)=e^x give you the same function. 

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