Derivative of 3 functions

Calculus Level pending

Given f ( x ) = a x m f(x) = ax^{m} , g ( x ) = b x n g(x) = bx^{n} , and h ( x ) = c x y h(x) = cx^{y} . The m-th, n-th, and y-th derivatives of f(x), g(x) and h(x) equals to 480, 30, and 30240 respectively. Given f(1) = 4, g(1) = 5 , and h(1) = 6, calculate m + n + y m n y \frac{m+n+y}{mny} . Round to nearest thousandths.


The answer is 0.142.

Jovan Alfian Djaja by Jovan Alfian Djaja , 18, Indonesia

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1 solution

If the degree of the derivative equals to the degree of the exponent, then it will produce a factorial. For instance, the m-th derivative of f(x) will equal a m ! am! , the n-th derivative of g(x) will equal b m ! bm! , and the y-th derivative of h(x) will equal c y ! cy! .Input the value of x = 1 to get f(1) = a = 4, g(1) = b = 5, and h(1) = c = 6. Input the value of a,b, and c to get m = 5, n = 3, and y = 7. Input the value of m,n,y to get 0.142. ( 5 + 3 + 7 5.3.7 \frac{5+3+7}{5 . 3 . 7} = 15 15.7 \frac{15}{15 . 7} = 0.142....)

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