Questions: Find dy/dx by implicit differentiation. y cos(x) = 5x^2 + 4y^2

Transcript text: Find $\frac{d y}{d x}$ by implicit differentiation. \[ \frac{y \cos (x)=5 x^{2}+4 y^{2}}{d x}=\square \]

Solution

Solution Steps

Step 1: Identify the given equation

The given equation is $- 5 x^{2} - 4 y^{2}{\left(x \right)} + y{\left(x \right)} \cos{\left(x \right)}=0$.

Step 2: Differentiate both sides of the equation with respect to x

The differentiation result is $- 10 x - y{\left(x \right)} \sin{\left(x \right)} - 8 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} y{\left(x \right)}=0$.

Step 3: Rearrange the resulting equation to solve for dy/dx

Rearranging gives $\frac{10 x + y{\left(x \right)} \sin{\left(x \right)}}{- 8 y{\left(x \right)} + \cos{\left(x \right)}}$ as the expression for $\frac{dy}{dx}$.

Final Answer:

The derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{10 x + y{\left(x \right)} \sin{\left(x \right)}}{- 8 y{\left(x \right)} + \cos{\left(x \right)}}$.

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