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If
y^(2)-x^(2)-3=0 then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $y^{2}-x^{2}-3=0$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Evaluate an exponential function

Full solution

Q. If

y^{2}-x^{2}-3=0

then find

\frac{d y}{d x}

in terms of

x

and

y

.

\newline

Answer:

\frac{d y}{d x}=

Given Equation: We are given the equation $y^{2} - x^{2} - 3 = 0$ and we need to find the derivative of $y$ with respect to $x$ , denoted as $\frac{dy}{dx}$ . To do this, we will use implicit differentiation, which involves taking the derivative of both sides of the equation with respect to $x$ .
Differentiate $y^2$ : First, we differentiate $y^{2}$ with respect to $x$ . Since $y$ is a function of $x$ , we use the chain rule and get $2y \cdot \frac{dy}{dx}$ .
Differentiate $-x^2$ : Next, we differentiate $-x^{2}$ with respect to $x$ , which is straightforward and equals $-2x$ .
Differentiate constant: The derivative of the constant $-3$ with respect to $x$ is $0$ , since the derivative of any constant is $0$ .
Combine and set equal: Now we combine the derivatives from the previous steps and set the derivative of the entire left side of the equation equal to the derivative of the right side, which is $0$ . This gives us the equation $2y \cdot \frac{dy}{dx} - 2x = 0$ .
Isolate $(dy)/(dx)$ : We solve for $(dy)/(dx)$ by isolating it on one side of the equation. To do this, we add $2x$ to both sides to get $2y \cdot (dy)/(dx) = 2x$ .
Divide by $2y$ : Finally, we divide both sides of the equation by $2y$ to solve for $\frac{dy}{dx}$ . This gives us $\frac{dy}{dx} = \frac{2x}{2y}$ .
Simplify expression: We can simplify the expression $\frac{2x}{2y}$ by canceling out the $2$ s, which leaves us with $\frac{dy}{dx} = \frac{x}{y}$ .

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