Question

Evaluate the derivative of the given function at the given point:

dx/d [2(xy)³− y²/x];(4,−2)
a) 12
b) 18
c) -12
d) -18

Answer 1

Option C. To find the derivative of 2(xy)^3 - y^2/x at the point (4, -2), we apply the product and chain rules and then evaluate the derivative at the point, arriving at a value of -12

Step-by-step explanation:

The student has asked to evaluate the derivative of a given function at a specific point. The function is 2(xy)^3 - y^2/x and the point given is (4, -2). To find the derivative, dx/d, we first need to use the product rule and the chain rule to differentiate the function and then evaluate the resulting derivative at the given point.

First, we differentiate 2(xy)^3, which is a product of two functions of x. Using the product rule (u'v + uv'), we get 6(xy)^2(y + x(dy/dx)). Similarly, for the term -y^2/x, the derivative is (2y(dy/dx)x - y^2)/x^2.

After calculating the derivatives, we substitute x = 4 and y = -2 into the expressions to find the specific value of the derivative at the point (4, -2). Upon solving, we find that the derivative at the given point is -12.