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) on the given curve, we can use implicit differentiation
To find the value of dy/dx at the point (-2, 4) on the given curve, we can use implicit differentiation. The equation of the curve is given as f(x)g(y) = 17 – x – y.
First, let’s differentiate both sides of the equation with respect to x using the product rule for differentiation:
d/dx (f(x)g(y)) = d/dx (17 – x – y)
Applying the product rule on the left side, we have:
g(y) * f'(x) + f(x) * g'(y) * dy/dx = -1
Now we need to find the values of f(x), f'(x), g(y), and g'(y) at the point (-2, 4). Using the values in the table above, we have:
f(-2) = -4, f'(-2) = 2, g(4) = 2, g'(4) = -3
Substituting these values into the equation, we get:
2 * g(4) – 4 * g'(4) * dy/dx = -1
Plugging in the given values, we have:
2 * 2 – 4 * (-3) * dy/dx = -1
Simplifying further:
4 + 12 * dy/dx = -1
Subtracting 4 from both sides:
12 * dy/dx = -5
Dividing by 12:
dy/dx = -5/12
So, the value of dy/dx at the point (-2, 4) is -5/12.
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