Use implicit differentiation to find the slope of the tangent line to the graph of 18*x^(1/2)+5y^(1/2)+14 at(81/324,1)
we have
18√x + 5√y = 14
9/√x + 5/(2√y) y' = 0
You have x and y, so plug 'em in to find y'
To find the slope of the tangent line to the graph of the equation 18x^(1/2) + 5y^(1/2) + 14 at the point (81/324, 1), we need to use implicit differentiation.
Step 1: Start by differentiating both sides of the equation with respect to x. Treat y as a function of x and use the chain rule for differentiating.
d/dx (18x^(1/2)) + d/dx(5y^(1/2)) + d/dx(14) = d/dx(0)
Step 2: Simplify the derivatives. The derivative of a constant with respect to x is just 0, so the last term on the left side of the equation goes away.
9/x^(1/2) + (5/2)(1/y^(1/2)) * dy/dx = 0
Step 3: Now, isolate dy/dx. Move the first term to the right side of the equation and multiply through by y^(1/2) to get rid of the fraction.
(5/2)(1/y^(1/2)) * dy/dx = -9/x^(1/2)
dy/dx = (-9/x^(1/2)) / [(5/2)(1/y^(1/2))]
Step 4: Simplify the expression. To simplify, we can rewrite x^(1/2) as √x and y^(1/2) as √y.
dy/dx = (-9/√x) / [(5/2)(1/√y)]
dy/dx = -18/5 * (√y/√x)
Step 5: Now, substitute the x and y values from the given point (81/324, 1) into the equation to find the slope of the tangent line.
dy/dx = -18/5 * (√1/√(81/324))
dy/dx = -18/5 * (√1/√(81/324))
dy/dx = -18/5 * √(324/81)
dy/dx = -18/5 * (18/9)
dy/dx = -18/5 * 2
dy/dx = -36/5
Therefore, the slope of the tangent line to the graph of 18x^(1/2) + 5y^(1/2) + 14 at the point (81/324, 1) is -36/5.