Using implicit differentiation, find $\frac{dy}{dx}$ . $\newline$ $-7x^{2}y^{4}-5xy^{2}=4-4x$

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Q. Using implicit differentiation, find

\frac{dy}{dx}

\newline

-7x^{2}y^{4}-5xy^{2}=4-4x

Apply Implicit Differentiation: To find the derivative $\frac{dy}{dx}$ using implicit differentiation, we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ . This means we will apply the product rule to terms involving both $x$ and $y$ , and the chain rule to terms involving $y$ .
Differentiate $-7x^{2}y^{4}$ : Differentiate the term $-7x^{2}y^{4}$ with respect to $x$ . Using the product rule, we get the derivative of the first function times the second function plus the first function times the derivative of the second function. The derivative of $x^{2}$ is $2x$ , and the derivative of $y^{4}$ with respect to $x$ is $4y^{3}(\frac{dy}{dx})$ by the chain rule. $\newline$ So, the derivative of $-7x^{2}y^{4}$ is $-7[2xy^{4} + x^{2}\cdot 4y^{3}(\frac{dy}{dx})]$ .
Differentiate $-5xy^{2}$ : Differentiate the term $-5xy^{2}$ with respect to $x$ . Again, using the product rule, we get the derivative of the first function times the second function plus the first function times the derivative of the second function. The derivative of $x$ is $1$ , and the derivative of $y^{2}$ with respect to $x$ is $2y\frac{dy}{dx}$ by the chain rule. $\newline$ So, the derivative of $-5xy^{2}$ is $-5[y^{2} + x\cdot 2y\frac{dy}{dx}]$ .
Differentiate $4-4x$ : Differentiate the right side of the equation, $4-4x$ , with respect to $x$ . The derivative of a constant is $0$ , and the derivative of $-4x$ with respect to $x$ is $-4$ . $\newline$ So, the derivative of $4-4x$ is $0 - 4$ .
Combine and Simplify: Combine the derivatives from the previous steps to write the differentiated equation: $\newline$ $-7[2xy^{4} + x^{2}\cdot4y^{3}\left(\frac{dy}{dx}\right)] - 5[y^{2} + x\cdot2y\left(\frac{dy}{dx}\right)] = -4$ .
Isolate Terms: Simplify the equation by distributing the constants and combining like terms: $\newline$ $-14xy^{4} - 28x^{2}y^{3}\frac{dy}{dx} - 5y^{2} - 10xy\frac{dy}{dx} = -4$ .
Factor Out $(\frac{dy}{dx}):$ Isolate terms with $(\frac{dy}{dx})$ on one side and the rest on the other side: $\newline$ \(-28x^{ $2$ }y^{ $3$ }(\frac{dy}{dx}) - $10$ xy(\frac{dy}{dx}) = $4$ + $14$ xy^{ $4$ } + $5$ y^{ $2$ }.
Solve for $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ from the left side of the equation: $\newline$ $\frac{dy}{dx}(-28x^{2}y^{3} - 10xy) = 4 + 14xy^{4} + 5y^{2}$ .
Solve for $(\frac{dy}{dx}):</b> Factor out \$(\frac{dy}{dx})$ from the left side of the equation: $\newline$ $(\frac{dy}{dx})(-28x^{2}y^{3} - 10xy) = 4 + 14xy^{4} + 5y^{2}$ .Solve for $(\frac{dy}{dx})$ by dividing both sides of the equation by $(-28x^{2}y^{3} - 10xy)$ : $\newline$ $(\frac{dy}{dx}) = \frac{4 + 14xy^{4} + 5y^{2}}{-28x^{2}y^{3} - 10xy}$ .

Using implicit differentiation, find dydx\frac{dy}{dx}dxdy​.\newline−7x2y4−5xy2=4−4x-7x^{2}y^{4}-5xy^{2}=4-4x−7x2y4−5xy2=4−4x

Using implicit differentiation, find $\frac{dy}{dx}$ . $\newline$ $-7x^{2}y^{4}-5xy^{2}=4-4x$