Help!!!
determine the slope of the graph x^2=ln(xy) at point (1,e)
x^2 = ln(xy)
x^2 = lnx + lny
2x = 1/x + 1/y y'
y' = y(2x - 1/x)
y'(1) = e(2-1) = e
or, more explicitly,
ln(xy) = x^2
xy = e^(x^2)
y = e^(x^2)/x
y' = e^(x^2) (2x-1)/x^2
y'(1) = e^1 (2-1)/1 = e
Thank you!!!!!!!
To determine the slope of the graph at the point (1, e), we need to find the derivative of the equation x^2 = ln(xy) with respect to both x and y, and then evaluate it at the point (1, e).
Step 1: Taking the natural logarithm of both sides of the equation x^2 = ln(xy), we get:
ln(x^2) = ln(ln(xy))
Step 2: Applying the logarithmic property, we can write it as:
2 ln(x) = ln(ln(xy))
Step 3: Differentiating both sides of the equation with respect to x, we get:
(2/x) = (1/ln(xy)) * (dy/dx)
Step 4: We can solve this equation for dy/dx by isolating it:
dy/dx = (2/x) * ln(xy)
Step 5: Now, substituting x = 1 and y = e into the equation, we get:
dy/dx = (2/1) * ln(1*e)
= 2 * ln(e)
= 2 * 1
= 2
Therefore, the slope of the graph x^2 = ln(xy) at the point (1, e) is 2.
To determine the slope of the graph at a specific point, you need to find the derivative of the function and substitute the given coordinates into it.
In this case, we want to find the slope of the graph of the equation x^2 = ln(xy) at the point (1, e).
Step 1: Find the derivative of the equation with respect to x.
Start by separating the variables using logarithm rules. Rewrite the equation as ln(xy) = x^2.
Now, take the derivative of both sides with respect to x.
Differentiating ln(xy) with respect to x involves applying the product rule. Let's break it down step by step:
d/dx[ln(xy)] = 1/(xy) * (y + x*(dy/dx))
= (y + x*(dy/dx))/(xy)
= (y/x) + (dy/dx)
Step 2: Plug in the given point (1, e) into the derivative.
Substitute x = 1 and y = e into the derivative we found:
dy/dx = (y/x) + (dy/dx)
= (e/1) + (dy/dx)
= e + (dy/dx)
Step 3: Solve for dy/dx.
Rearrange the equation to solve for dy/dx:
dy/dx = e - (dy/dx)
2*(dy/dx) = e
dy/dx = e/2
Therefore, the slope of the graph of x^2 = ln(xy) at the point (1, e) is e/2.