The point (−2,4) lies on the curve in the xy-plane given by the equation f(x)g(y)=17−x−y, where f is a differentiable function of x and g is a differentiable function of y. Selected values of f, f′, g, and g′ are given in the table above. What is the value of dydx at the point (−2,4) ?
To find the value of dy/dx at the point (-2,4), we need to use implicit differentiation and the given equation f(x)g(y) = 17 – x – y
To find the value of dy/dx at the point (-2,4), we need to use implicit differentiation and the given equation f(x)g(y) = 17 – x – y.
First, let’s differentiate both sides of the equation with respect to x:
d/dx [f(x)g(y)] = d/dx [17 – x – y]
To differentiate f(x)g(y), we need to use the product rule. According to the product rule, the derivative of the product of two functions u(x) and v(x) is given by:
d/dx [u(x)v(x)] = u(x)v'(x) + v(x)u'(x)
Applying the product rule to f(x)g(y), we have:
f'(x)g(y) + f(x)g'(y) * dy/dx = -1
Now, we can substitute the values given in the table for f(-2), f'(-2), g(4), and g'(4) into the equation:
f'(-2) * g(4) + f(-2) * g'(4) * dy/dx = -1
Since we are trying to find dy/dx at the point (-2,4), we can substitute f(-2) = 1, g(4) = 2, f'(-2) = -1, and g'(4) = 3 into the equation:
-1 * 2 + 1 * 3 * dy/dx = -1
Simplifying the equation:
-2 + 3dy/dx = -1
3dy/dx = 1 – (-2)
3dy/dx = 3
Dividing both sides by 3:
dy/dx = 1
Therefore, the value of dy/dx at the point (-2,4) is 1.
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