If -3+4x^(2)=4y^(3)+3y+5y^(2) then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Differentiate with respect to x: First, we need to differentiate both sides of the equation with respect to x. The equation is -3 + 4x^2 = 4y^3 + 3y + 5y^2. We will apply the derivative to each term separately, remembering that y is a function of x, so we will use the chain rule for those terms.Left side derivative: Differentiate the left side of the equation with respect to x: d/dx(-3 + 4x^2). The derivative of a constant is 0, and the derivative of 4x^2 with respect to x is 8x. So, d/dx(-3 + 4x^2) = 0 + 8x = 8x.Right side derivative: Differentiate the right side of the equation with respect to x: d/dx(4y^3 + 3y + 5y^2). Using the chain rule, the derivative of 4y^3 with respect to x is 12y^2(dy/dx), the derivative of 3y with respect to x is 3(dy/dx), and the derivative of 5y^2 with respect to x is 10y(dy/dx). So, d/dx(4y^3 + 3y + 5y^2) = 12y^2(dy/dx) + 3(dy/dx) + 10y(dy/dx).Equate derivatives: Now we equate the derivatives from both sides: 8x = 12y^2(dy/dx) + 3(dy/dx) + 10y(dy/dx).Solve for dy/dx: We need to solve for (dy/dx). To do this, we factor out (dy/dx) from the right side of the equation: 8x = (dy/dx)(12y^2 + 3 + 10y).Isolate dy/dx: Now, divide both sides by (12y^2 + 3 + 10y) to isolate (dy/dx): (dy/dx) = 8x / (12y^2 + 3 + 10y).

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If
-3+4x^(2)=4y^(3)+3y+5y^(2) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $-3+4 x^{2}=4 y^{3}+3 y+5 y^{2}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

View step-by-step help

Full solution

Q. If

-3+4 x^{2}=4 y^{3}+3 y+5 y^{2}

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Differentiate with respect to $x$ : First, we need to differentiate both sides of the equation with respect to $x$ . The equation is $-3 + 4x^2 = 4y^3 + 3y + 5y^2$ . We will apply the derivative to each term separately, remembering that $y$ is a function of $x$ , so we will use the chain rule for those terms.
Left side derivative: Differentiate the left side of the equation with respect to $x$ : $\frac{d}{dx}(-3 + 4x^2)$ . The derivative of a constant is $0$ , and the derivative of $4x^2$ with respect to $x$ is $8x$ . $\newline$ So, $\frac{d}{dx}(-3 + 4x^2) = 0 + 8x = 8x$ .
Right side derivative: Differentiate the right side of the equation with respect to $x$ : $\frac{d}{dx}(4y^3 + 3y + 5y^2)$ . Using the chain rule, the derivative of $4y^3$ with respect to $x$ is $12y^2\frac{dy}{dx}$ , the derivative of $3y$ with respect to $x$ is $3\frac{dy}{dx}$ , and the derivative of $5y^2$ with respect to $x$ is $\frac{d}{dx}(4y^3 + 3y + 5y^2)$ $0$ . So, $\frac{d}{dx}(4y^3 + 3y + 5y^2)$ $1$ .
Equate derivatives: Now we equate the derivatives from both sides: $8x = 12y^2\frac{dy}{dx} + 3\frac{dy}{dx} + 10y\frac{dy}{dx}$ .
Solve for $\frac{dy}{dx}$ : We need to solve for $\frac{dy}{dx}$ . To do this, we factor out $\frac{dy}{dx}$ from the right side of the equation: $8x = \left(\frac{dy}{dx}\right)(12y^2 + 3 + 10y)$ .
Isolate $\frac{dy}{dx}$ : Now, divide both sides by $(12y^2 + 3 + 10y)$ to isolate $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{8x}{12y^2 + 3 + 10y}$ .