Question Use implicit differentiation to find the derivative dy/dx. Find the slope of the curve at the point (9,1) for the equation $2xy + 5x^(3/2)y^(-1/2) = 153$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $2xy + 5x^{3/2}y^{-1/2} = 153$ .
2. We need to use implicit differentiation to find $\frac{dy}{dx}$ .
3. We are given a specific point on the curve, $(9,1)$ , to find the slope at that point.

STEP 2

Begin by applying implicit differentiation to both sides of the equation with respect to $x$ . Remember that $y$ is a function of $x$ , so when differentiating terms with $y$ , we need to use the chain rule.
$\frac{d}{dx}(2xy) + \frac{d}{dx}(5x^{3/2}y^{-1/2}) = \frac{d}{dx}(153)$

STEP 3

Differentiate $2xy$ using the product rule, which states that $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x$ and $v = y$ .
$\frac{d}{dx}(2xy) = 2\frac{d}{dx}(x)y + 2x\frac{d}{dx}(y) = 2y + 2x\frac{dy}{dx}$

STEP 4

Differentiate $5x^{3/2}y^{-1/2}$ using the product rule. Let $u = 5x^{3/2}$ and $v = y^{-1/2}$ .
$\frac{d}{dx}(5x^{3/2}y^{-1/2}) = \frac{d}{dx}(5x^{3/2})y^{-1/2} + 5x^{3/2}\frac{d}{dx}(y^{-1/2})$

STEP 5

Differentiate $5x^{3/2}$ with respect to $x$ .
$\frac{d}{dx}(5x^{3/2}) = 5 \cdot \frac{3}{2}x^{3/2 - 1} = \frac{15}{2}x^{1/2}$

STEP 6

Differentiate $y^{-1/2}$ with respect to $x$ using the chain rule.
$\frac{d}{dx}(y^{-1/2}) = -\frac{1}{2}y^{-1/2 - 1}\frac{dy}{dx} = -\frac{1}{2}y^{-3/2}\frac{dy}{dx}$

STEP 7

Combine the results from STEP_5 and STEP_6.
$\frac{d}{dx}(5x^{3/2}y^{-1/2}) = \frac{15}{2}x^{1/2}y^{-1/2} - \frac{5}{2}x^{3/2}y^{-3/2}\frac{dy}{dx}$

STEP 8

Combine the differentiated terms from STEP_3 and STEP_7 and set the derivative of the constant 153 to 0.
$2y + 2x\frac{dy}{dx} + \frac{15}{2}x^{1/2}y^{-1/2} - \frac{5}{2}x^{3/2}y^{-3/2}\frac{dy}{dx} = 0$

STEP 9

Isolate the terms with $\frac{dy}{dx}$ on one side of the equation.
$2x\frac{dy}{dx} - \frac{5}{2}x^{3/2}y^{-3/2}\frac{dy}{dx} = -2y - \frac{15}{2}x^{1/2}y^{-1/2}$

STEP 10

Factor out $\frac{dy}{dx}$ from the left side of the equation.
$\frac{dy}{dx}(2x - \frac{5}{2}x^{3/2}y^{-3/2}) = -2y - \frac{15}{2}x^{1/2}y^{-1/2}$

STEP 11

Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{-2y - \frac{15}{2}x^{1/2}y^{-1/2}}{2x - \frac{5}{2}x^{3/2}y^{-3/2}}$

STEP 12

Now we need to find the slope of the curve at the given point $(9,1)$ . Substitute $x = 9$ and $y = 1$ into the expression for $\frac{dy}{dx}$ .
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{-2(1) - \frac{15}{2}(9)^{1/2}(1)^{-1/2}}{2(9) - \frac{5}{2}(9)^{3/2}(1)^{-3/2}}$

STEP 13

Simplify the expression by evaluating the powers of 9 and 1.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{-2 - \frac{15}{2}(3)}{18 - \frac{5}{2}(27)}$

STEP 14

Perform the arithmetic operations.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{-2 - \frac{45}{2}}{18 - \frac{135}{2}}$

STEP 15

Combine the terms in the numerator and the denominator.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{-2 - 22.5}{18 - 67.5}$

STEP 16

Simplify the fractions.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{-24.5}{-49.5}$

STEP 17

Reduce the fraction to its simplest form.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{24.5}{49.5}$

STEP 18

Divide both numerator and denominator by 24.5 to get the slope.
$\frac{dy}{dx}\bigg|_{(9,1)} = \frac{1}{2}$
The slope of the curve at the point $(9,1)$ is $\frac{1}{2}$ .

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