Answer :
The function f(x) =[tex]25x^2 + 25x^3 + 25x^4[/tex] can be derived by applying the power rule of differentiation to each term individually and then summing the derivatives.
To find the derivative of the given function f(x) = [tex]25x^2 + 25x^3 + 25x^4[/tex], we can apply the power rule of differentiation. According to the power rule, the derivative of x^n, where n is a constant, is given by n*x^(n-1).
Taking the derivative of each term individually, we have:
[tex]d/dx (25x^2) = 2*25*x^(2-1) = 50x\\d/dx (25x^3) = 3*25*x^(3-1) = 75x^2\\d/dx (25x^4) = 4*25*x^(4-1) = 100x^3\\[/tex]
Now, we can sum up the derivatives of each term to find the derivative of the entire function:f'(x) = d/dx
[tex](25x^2) + d/dx (25x^3) + d/dx (25x^4)\\ = 50x + 75x^2 + 100x^3[/tex]
Thus, the derivative of f(x) = [tex]25x^2 + 25x^3 + 25x^4 is f'(x) = 50x + 75x^2 + 100x^3.[/tex]
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