If 5x^(2)-5-3y^(2)=0 then find (dy)/(dx) at the point (4,5). Answer: (dy)/(dx)|_((4,5))=

Implicit Differentiation: To find (dy)/(dx), we need to differentiate the given equation implicitly with respect to x. The given equation is 5x^2 - 5 - 3y^2 = 0. Differentiating both sides with respect to x, we get: d/dx(5x^2) - d/dx(5) - d/dx(3y^2) = d/dx(0)Derivative Calculation: Differentiating each term separately, we have: d/dx(5x^2) = 10x (since the derivative of x^2 with respect to x is 2x, and we multiply by the constant 5), d/dx(5) = 0 (since the derivative of a constant is 0), d/dx(3y^2) = 6y(dy/dx) (since we are differentiating with respect to x, y is treated as a function of x, so we use the chain rule). So, we have 10x - 0 - 6y(dy/dx) = 0.Solving for dy/dx: Now we solve for dy/dx: 10x - 6y(dy/dx) = 0 6y(dy/dx) = 10x (dy/dx) = 10x / (6y)Substitution of Given Point: We substitute the given point (4,5) into the equation to find the value of dy/dx at that point: (dy/dx)|_((4,5)) = 10(4) / (6(5))Final Calculation: Perform the calculation: (dy/dx)|_((4,5)) = 40 / 30 (dy/dx)|_((4,5)) = 4 / 3

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Math Problems

Grade 8

Evaluate a nonlinear function

Full solution

Q. If

5 x^{2}-5-3 y^{2}=0

then find

\frac{d y}{d x}

at the point

(4,5)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(4,5)}=

Implicit Differentiation: To find $\frac{dy}{dx}$ , we need to differentiate the given equation implicitly with respect to $x$ . The given equation is $5x^2 - 5 - 3y^2 = 0$ . Differentiating both sides with respect to $x$ , we get: \frac{d}{dx}(\(5x^ $2$ ) - \frac{d}{dx}( $5$ ) - \frac{d}{dx}( $3$ y^ $2$ ) = \frac{d}{dx}( $0$ )
Derivative Calculation: Differentiating each term separately, we have: $\newline$ $\frac{d}{dx}(5x^2) = 10x$ (since the derivative of $x^2$ with respect to $x$ is $2x$ , and we multiply by the constant $5$ ), $\newline$ $\frac{d}{dx}(5) = 0$ (since the derivative of a constant is $0$ ), $\newline$ $\frac{d}{dx}(3y^2) = 6y\frac{dy}{dx}$ (since we are differentiating with respect to $x$ , $y$ is treated as a function of $x$ , so we use the chain rule). $\newline$ So, we have $x^2$ $1$ .
Solving for $\frac{dy}{dx}$ : Now we solve for $\frac{dy}{dx}$ : $10x - 6y\left(\frac{dy}{dx}\right) = 0$ $6y\left(\frac{dy}{dx}\right) = 10x$ $\frac{dy}{dx} = \frac{10x}{6y}$
Substitution of Given Point: We substitute the given point $(4,5)$ into the equation to find the value of $\frac{dy}{dx}$ at that point: $\newline$ $\left(\frac{dy}{dx}\right)|_{(4,5)} = \frac{10(4)}{6(5)}$
Final Calculation: Perform the calculation: $\newline$ $(\frac{dy}{dx})|_{(4,5)} = \frac{40}{30}$ $\newline$ $(\frac{dy}{dx})|_{(4,5)} = \frac{4}{3}$