use implicit differentiation to find dy/dx . Then evaluate the derivative function at the designated point.
log (xy)= 2; (4, 25)
log(xy) = 2
1/(xy) (y + xy') = 0
1/x + 1/y y' = 0
y' = -y/x
Or, going about it another way,
logx + logy = 2
1/x + 1/y y' = 0
y' = -y/x
So, plug in your numbers.
To find the derivative of the function using implicit differentiation, we need to differentiate both sides of the equation with respect to x.
The given equation is log(xy) = 2.
Step 1: Differentiate both sides of the equation with respect to x.
For the left-hand side, we need to use the chain rule:
d/dx [log(xy)] = (1/xy) * (y + x(dy/dx))
For the right-hand side, we differentiate the constant term '2', which will be zero.
So, applying the chain rule and differentiating both sides, we get:
(1/xy) * (y + x(dy/dx)) = 0
Step 2: Simplify the equation.
Let's rearrange the equation:
y + xy(dy/dx) = 0
Step 3: Solve for dy/dx.
Now, isolate the term dy/dx by moving the y term to the other side:
xy(dy/dx) = -y
Finally, divide both sides by xy to solve for dy/dx:
(dy/dx) = -y / xy
Step 4: Evaluate the derivative function at the given point (4, 25).
Substitute x = 4 and y = 25 into the derivative function:
(dy/dx) = -(25) / (4 * 25)
Simplifying further:
(dy/dx) = -1/4
So, the derivative of the function is dy/dx = -1/4 at the point (4, 25).