Questions: Find (d y / d x) by differentiating implicitly. When applicable, express the result in terms of (x) and (y) (4(x^2+1)^5+(y^2+2)^2=27) (fracd yd x=square)

$Find (d y / d x) by differentiating implicitly. When applicable, express the result in terms of (x) and (y) (4(x^2+1)^5+(y^2+2)^2=27) (fracd yd x=square)$

Transcript text: Find $d y / d x$ by differentiating implicitly. When applicable, express the result in terms of $x$ and $y$ \[ \begin{array}{l} 4\left(x^{2}+1\right)^{5}+\left(y^{2}+2\right)^{2}=27 \\ \frac{d y}{d x}=\square \end{array} \] $\square$

Solution

Solution Steps

To find $ \frac{dy}{dx} $ by differentiating implicitly, we will differentiate both sides of the given equation with respect to $ x $. We will apply the chain rule to differentiate the composite functions and the power rule where applicable. After differentiating, we will solve for $ \frac{dy}{dx} $ by isolating it on one side of the equation.

Step 1: Differentiate the Equation

We start with the equation: \[ 4\left(x^{2}+1\right)^{5}+\left(y^{2}+2\right)^{2}=27 \] To find $ \frac{dy}{dx} $, we differentiate both sides with respect to $ x $. Using the chain rule, we differentiate the left-hand side: \[ \frac{d}{dx}\left(4\left(x^{2}+1\right)^{5}\right) + \frac{d}{dx}\left(\left(y^{2}+2\right)^{2}\right) = 0 \] This gives us: \[ 40x\left(x^{2}+1\right)^{4} + 2\left(y^{2}+2\right)\cdot 2y\frac{dy}{dx} = 0 \]

Step 2: Isolate $ \frac{dy}{dx} $

Rearranging the differentiated equation to isolate $ \frac{dy}{dx} $: \[ 2\left(y^{2}+2\right)\cdot y\frac{dy}{dx} = -40x\left(x^{2}+1\right)^{4} \] Now, we can solve for $ \frac{dy}{dx} $: \[ \frac{dy}{dx} = -\frac{40x\left(x^{2}+1\right)^{4}}{2y\left(y^{2}+2\right)} = -\frac{20x\left(x^{2}+1\right)^{4}}{y\left(y^{2}+2\right)} \]