If -5y+3y^(2)+4+5x^(3)=x^(2) then find (dy)/(dx) at the point (-1,2). Answer: (dy)/(dx)|_((-1,2))=

Implicit Differentiation: First, we need to implicitly differentiate both sides of the equation with respect to x. The equation is -5y + 3y^2 + 4 + 5x^3 = x^2.Differentiation with Chain Rule: Differentiate each term with respect to x. For terms involving y, we use the chain rule, treating y as a function of x (y = y(x)). d/dx(-5y) = -5(dy/dx), since y is a function of x. d/dx(3y^2) = 3 * 2y * (dy/dx) = 6y(dy/dx), again using the chain rule. d/dx(4) = 0, since the derivative of a constant is zero. d/dx(5x^3) = 5 * 3x^2 = 15x^2, using the power rule. d/dx(x^2) = 2x, using the power rule.Write Differentiated Equation: Now, we write down the differentiated equation: -5(dy/dx) + 6y(dy/dx) + 0 + 15x^2 = 2x.Simplify Combined Terms: We can simplify the equation by combining like terms: (6y - 5)(dy/dx) + 15x^2 = 2x.Solve for dy/dx: Now, we solve for dy/dx: (dy/dx) = (2x - 15x^2) / (6y - 5).Substitute Values: We substitute x = -1 and y = 2 into the equation to find dy/dx at the point (-1,2): (dy/dx)|_((-1,2)) = (2(-1) - 15(-1)^2) / (6(2) - 5).Calculate Result: Calculate the values: (dy/dx)|_((-1,2)) = (-2 - 15) / (12 - 5).Final Simplification: Simplify the expression: (dy/dx)|_((-1,2)) = (-17) / 7.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If
-5y+3y^(2)+4+5x^(3)=x^(2) then find
(dy)/(dx) at the point
(-1,2).
Answer:
(dy)/(dx)|_((-1,2))=

If $-5 y+3 y^{2}+4+5 x^{3}=x^{2}$ then find $\frac{d y}{d x}$ at the point $(-1,2)$ . $\newline$ Answer: $\left.\frac{d y}{d x}\right|_{(-1,2)}=$

View step-by-step help

Home

Math Problems

Algebra 2

Transformations of absolute value functions

Full solution

Q. If

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

then find

\frac{d y}{d x}

at the point

(-1,2)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(-1,2)}=

Implicit Differentiation: First, we need to implicitly differentiate both sides of the equation with respect to $x$ . The equation is $-5y + 3y^2 + 4 + 5x^3 = x^2$ .
Differentiation with Chain Rule: Differentiate each term with respect to $x$ . For terms involving $y$ , we use the chain rule, treating $y$ as a function of $x$ ( $y = y(x)$ ). $\newline$ $\frac{d}{dx}(-5y) = -5\frac{dy}{dx}$ , since $y$ is a function of $x$ . $\newline$ $\frac{d}{dx}(3y^2) = 3 \cdot 2y \cdot \frac{dy}{dx} = 6y\frac{dy}{dx}$ , again using the chain rule. $\newline$ $\frac{d}{dx}(4) = 0$ , since the derivative of a constant is zero. $\newline$ $y$ $0$ , using the power rule. $\newline$ $y$ $1$ , using the power rule.
Write Differentiated Equation: Now, we write down the differentiated equation: $\newline$ $-5\frac{dy}{dx} + 6y\frac{dy}{dx} + 0 + 15x^2 = 2x$ .
Simplify Combined Terms: We can simplify the equation by combining like terms: $(6y - 5)\frac{dy}{dx} + 15x^2 = 2x.$
Solve for $\frac{dy}{dx}$ : Now, we solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{2x - 15x^2}{6y - 5}$ .
Substitute Values: We substitute $x = -1$ and $y = 2$ into the equation to find $\frac{dy}{dx}$ at the point $(-1,2)$ : $\left(\frac{dy}{dx}\right)|_{(-1,2)} = \frac{(2(-1) - 15(-1)^2)}{(6(2) - 5)}$ .
Calculate Result: Calculate the values: $\newline$ $(\frac{dy}{dx})|_{(-1,2)} = \frac{(-2 - 15)}{(12 - 5)}.$
Final Simplification: Simplify the expression: $(\frac{dy}{dx})|_{(\(-1$,$2$)} = \frac{$-17$}{$7$}\.