QuestionFind $\frac{d y}{d x}$ using implicit differentiation for $xy=35$ at the point $(-5,-7)$ . What is the slope?

Studdy Solution

STEP 1

Assumptions1. The given equation is $xy =35$ . We are asked to find $\frac{dy}{dx}$ by implicit differentiation.
3. We are also asked to find the slope of the graph at the point $(-5,-7)$ .

STEP 2

First, we need to differentiate the given equation with respect to $x$ . The equation is $xy =35$ . We can use the product rule of differentiation, which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first.
$\frac{d}{dx}(xy) = x \frac{dy}{dx} + y \frac{dx}{dx}$

STEP 3

Now, differentiate the right side of the equation with respect to $x$ . The derivative of a constant is zero.
$\frac{d}{dx}(35) =0$

STEP 4

Now, we can set the derivatives of both sides of the equation equal to each other.
$x \frac{dy}{dx} + y \frac{dx}{dx} =0$

STEP 5

The derivative of $x$ with respect to $x$ is1, so we can simplify the equation.
$x \frac{dy}{dx} + y =0$

STEP 6

Now, solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = -\frac{y}{x}$

STEP 7

Now that we have $\frac{dy}{dx}$ , we can find the slope of the graph at the point $(-5,-7)$ by substituting these values into the equation.
$\frac{dy}{dx} = -\frac{-7}{-5}$

STEP 8

Calculate the slope at the given point.
$\frac{dy}{dx} = -\frac{-7}{-5} = \frac{7}{5}$ The derivative of $y$ with respect to $x$ is $-\frac{y}{x}$ , and the slope of the graph at the point $(-5,-7)$ is $\frac{7}{5}$ .

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