Final answer:
To solve the differential equation at the point (-4, -9), implicit differentiation of the equation xy = 36 gives us y' = -y/x. Substituting the given point into this equation, we find that the slope of the tangent line, y', is 9/4 at (-4, -9).
Step-by-step explanation:
The question asks us to solve a differential equation at the given point (-4, -9). This is a calculus problem where 'y' is the dependent variable and 'y'' denotes the derivative of 'y' with respect to 'x', the independent variable. The equation xy = 36 suggests a hyperbola, and we need to find the slope of the tangent line to this curve at the point (-4, -9), which is denoted by 'y''.
To find the derivative 'y'', we differentiate both sides of the equation xy = 36 with respect to 'x'. Using implicit differentiation, we get:
x(dy/dx) + y*(1) = 0
Which simplifies to:
dy/dx = -y/x
At the point (-4, -9), we substitute 'y' with -9 and 'x' with -4:
y' = -(-9)/(-4)
Therefore, y' = 9/4.
The slope of the tangent line to the hyperbola at the point (-4, -9) is 9/4.