Find the slope of the graph of the function at the given point. (If an answer is undefined, enter UNDEFINED.)
x^3 + y^3 = 14xy, (7, 7)
find dy/dx at (7.7)
3x^2 dx+3y^2 dy=14(ydx+xdy)
dy/dx(3y^2-14x)=14y-3x^2
check that, I did the algebra in my head.
dy/dx= 108-147 / 147-98
check that, it was in my head also.
To find the slope of the graph of the function at a given point, you need to take the derivative of the function and evaluate it at the given point.
Let's start by differentiating the given equation with respect to x:
d/dx (x^3 + y^3) = d/dx (14xy)
To find d/dx (y^3), we need to use the chain rule. Since y is a function of x, we can differentiate it as follows:
d/dx (y^3) = d/dy (y^3) * dy/dx
Now, let's find the derivative of x^3:
d/dx (x^3) = 3x^2
Next, let's find the derivative of y^3:
d/dy (y^3) = 3y^2
To find dy/dx, we can rearrange the original equation and solve for dy/dx:
x^3 + y^3 = 14xy
Differentiating both sides with respect to x:
3x^2 + 3y^2 * dy/dx = 14y + 14x * dy/dx
Rearranging terms:
3x^2 - 14x * dy/dx = 14y - 3y^2 * dy/dx
Simplifying:
dy/dx * (3y^2 + 14x) = 14y - 3x^2
dy/dx = (14y - 3x^2) / (3y^2 + 14x)
Now, to find the slope at the given point (7, 7), substitute x = 7 and y = 7 into the derivative equation:
dy/dx = (14(7) - 3(7)^2) / (3(7)^2 + 14(7))
dy/dx = (98 - 147) / (147 + 98)
dy/dx = -49 / 245
dy/dx = -1/5
Therefore, the slope of the graph of the function at the point (7, 7) is -1/5.