USE IMPLICIT $\frac{d\varphi}{dx}$ to find $\frac{d\varphi}{dx}$ for $-7xY-5$

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Simplify radical expressions with variables II

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Q. USE IMPLICIT

\frac{d\varphi}{dx}

to find

\frac{d\varphi}{dx}

for

-7xY-5

Given expression and differentiation: We are given the expression $-7x^2y - 5$ and we need to find $\frac{d\varphi}{dx}$ using implicit differentiation. Since the expression does not explicitly solve for $y$ in terms of $x$ , we differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (i.e., $y(x)$ ).
Product rule application: Differentiate $-7x^2y$ with respect to $x$ . Using the product rule, which states that $d(uv)/dx = u(dv/dx) + v(du/dx)$ , we differentiate $-7x^2$ and $y$ separately. The derivative of $-7x^2$ with respect to $x$ is $-14x$ , and the derivative of $y$ with respect to $x$ is $x$ $0$ since $y$ is a function of $x$ .
Derivative of constant: Applying the product rule, we get $-7x^2 \cdot \frac{dy}{dx} - 14xy$ as the derivative of $-7x^2y$ with respect to $x$ .
Combining derivatives: Differentiate $-5$ with respect to $x$ . The derivative of a constant is $0$ , so the derivative of $-5$ with respect to $x$ is $0$ .
Equating to find derivative: Combine the derivatives from the previous steps to write the full derivative of the expression $-7x^2y - 5$ with respect to $x$ . The full derivative is $-7x^2 \cdot \frac{dy}{dx} - 14xy + 0$ .
Isolating and solving: Since we are looking for $\frac{d\varphi}{dx}$ , and $\varphi$ represents the expression $-7x^2y - 5$ , we can equate the derivative to $0$ because the expression is not equal to a function of $x$ , but rather a constant (which we can assume to be $0$ for the purpose of finding the derivative). $\newline$ So, we have $-7x^2 \cdot \frac{dy}{dx} - 14xy = 0$ .
Simplifying the expression: Solve for $\frac{dy}{dx}$ . We isolate $\frac{dy}{dx}$ on one side of the equation to find its value. $\newline$ $-7x^2 \cdot \frac{dy}{dx} = 14xy$ $\newline$ $\frac{dy}{dx} = \frac{14xy}{-7x^2}$
Simplifying the expression: Solve for $\frac{dy}{dx}$ . We isolate $\frac{dy}{dx}$ on one side of the equation to find its value. $\newline$ $-7x^2 \cdot \frac{dy}{dx} = 14xy$ $\newline$ $\frac{dy}{dx} = \frac{14xy}{-7x^2}$ Simplify the expression for $\frac{dy}{dx}$ . We can cancel out a $x$ from the numerator and denominator, and also simplify the constants. $\newline$ $\frac{dy}{dx} = \frac{14y}{-7x}$ $\newline$ $\frac{dy}{dx} = -2\frac{y}{x}$