Final answer:
To find the derivative dy/dx at x = 3 for the given function y = 5x/√(x^2 + 1), we apply the quotient rule to differentiate the function with respect to x and then substitute x with 3 in the simplified derivative expression.
Step-by-step explanation:
The question involves finding the derivative of a given function with respect to 'x' at a specific point (x=3). This problem falls under the category of differential calculus, a fundamental topic of high school and college mathematics that deals with rates of change. The function provided is y = 5x/√(x^2 + 1).
To find the derivative, dy/dx, we need to apply the quotient rule since we have a function of x divided by another function of x. The quotient rule states that if we have a function h(x) = f(x)/g(x), then its derivative h'(x) is given by h'(x) = (g(x)f'(x) - f(x)g'(x))/(g(x))^2.
Therefore, for y = 5x/√(x^2 + 1), we let f(x) = 5x and g(x) = √(x^2 + 1). The derivative of f(x) with respect to x is f'(x) = 5 and the derivative of g(x) is g'(x) = x/√(x^2 + 1) by chain rule. Applying the quotient rule, we get:
dy/dx = [√(x^2 + 1)⋅(5) - 5x⋅(x/√(x^2 + 1))]/(x^2 + 1)
Simplifying the expression, we find the derivative for all values of x. To find dy/dx at x = 3, we substitute x with 3 in the derivative expression and simplify to get the required value.