If -y+x+y^(2)+5y^(3)+3x^(2)=0 then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Given Equation: We are given the equation -y + x + y^(2) + 5y^(3) + 3x^(2) = 0 and we need to find the derivative of y with respect to x, which is (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, while treating y as a function of x.Implicit Differentiation: First, we differentiate each term of the equation with respect to x. For the terms involving only x, we use the standard rules of differentiation. For the terms involving y, we use the chain rule, remembering to multiply by (dy)/(dx) since y is a function of x. The derivative of -y with respect to x is -1(dy)/(dx). The derivative of x with respect to x is 1. The derivative of y^(2) with respect to x is 2y(dy)/(dx). The derivative of 5y^(3) with respect to x is 15y^(2)(dy)/(dx). The derivative of 3x^(2) with respect to x is 6x.Differentiating Terms: Now we write down the differentiated equation: -1(dy)/(dx) + 1 + 2y(dy)/(dx) + 15y^(2)(dy)/(dx) + 6x = 0Writing Differentiated Equation: Next, we collect all the terms involving (dy)/(dx) on one side and the terms not involving (dy)/(dx) on the other side: -1(dy)/(dx) + 2y(dy)/(dx) + 15y^(2)(dy)/(dx) = -1 - 6xSolving for (dy)/(dx): We factor out (dy)/(dx) from the left side of the equation: (dy)/(dx)(-1 + 2y + 15y^(2)) = -1 - 6xNow we solve for (dy)/(dx) by dividing both sides of the equation by (-1 + 2y + 15y^(2)): (dy)/(dx) = (-1 - 6x) / (-1 + 2y + 15y^(2))

Home

Math Problems

Algebra 2

Evaluate exponential functions

Full solution

Q. If

-y+x+y^{2}+5 y^{3}+3 x^{2}=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 $-y + x + y^{2} + 5y^{3} + 3x^{2} = 0$ and we need to find the derivative of $y$ with respect to $x$ , which is $\frac{dy}{dx}$ . To do this, we will use implicit differentiation, which involves taking the derivative of both sides of the equation with respect to $x$ , while treating $y$ as a function of $x$ .
Implicit Differentiation: First, we differentiate each term of the equation with respect to $x$ . For the terms involving only $x$ , we use the standard rules of differentiation. For the terms involving $y$ , we use the chain rule, remembering to multiply by $\frac{dy}{dx}$ since $y$ is a function of $x$ . The derivative of $-y$ with respect to $x$ is $-1\frac{dy}{dx}$ . The derivative of $x$ with respect to $x$ is $x$ $1$ . The derivative of $x$ $2$ with respect to $x$ is $x$ $4$ . The derivative of $x$ $5$ with respect to $x$ is $x$ $7$ . The derivative of $x$ $8$ with respect to $x$ is $y$ $0$ .
Differentiating Terms: Now we write down the differentiated equation: $\newline$ $-1\frac{dy}{dx} + 1 + 2y\frac{dy}{dx} + 15y^{2}\frac{dy}{dx} + 6x = 0$
Writing Differentiated Equation: Next, we collect all the terms involving $\frac{dy}{dx}$ on one side and the terms not involving $\frac{dy}{dx}$ on the other side: $\newline$ \(-1\frac{dy}{dx} + $2$ y\frac{dy}{dx} + $15$ y^{ $2$ }\frac{dy}{dx} = $-1$ - $6$ x
Solving for $(\frac{dy}{dx}):</b> We factor out \$(\frac{dy}{dx})$ from the left side of the equation: $\newline$ (\frac{dy}{dx})(\(-1 + $2$ y + $15$ y^{ $2$ }) = $-1$ - $6$ x
Solving for $(\frac{dy}{dx}):</b> We factor out \$(\frac{dy}{dx})$ from the left side of the equation: $\newline$ $(\frac{dy}{dx})(-1 + 2y + 15y^{2}) = -1 - 6x$ Now we solve for $(\frac{dy}{dx})$ by dividing both sides of the equation by $(-1 + 2y + 15y^{2}):$ $\newline$ $(\frac{dy}{dx}) = \frac{-1 - 6x}{-1 + 2y + 15y^{2}}$

More problems from Evaluate exponential functions

Question
Solve for $x$ . $\newline$ $5^x=1$ $\newline$ $\newline$ Write your answer in simplest form.
Get tutor help
Posted 5 months ago

Question
Steven opened a savings account and deposited $\$100.00$ as principal. The account earns $9\%$ interest, compounded annually. What is the balance after $5$ years? $\newline$ Use the formula $A = P(1 + \frac{r}{n})^{nt}$ , where $A$ is the balance (final amount), $P$ is the principal (starting amount), $r$ is the interest rate expressed as a decimal, $n$ is the number of times per year that the interest is compounded, and $t$ is the time in years. $\newline$ Round your answer to the nearest cent.
Get tutor help
Posted 4 months ago

Question
Use the following function rule to find $f(1)$ . $\newline$ $f(x) = 12(7)^x + 8$
Get tutor help
Posted 5 months ago

Question
The town of Valley View conducted a census this year, which showed that it has a population of $1,800$ people. Based on the census data, it is estimated that the population of Valley View will grow by $12\%$ each decade. $\newline$ Write an exponential equation in the form $y = a(b)^x$ that can model the town population, $y$ , $x$ decades after this census was taken. $\newline$ Use whole numbers, decimals, or simplified fractions for the values of $a$ and $b$ . $\newline$ $y =$ ______
Get tutor help
Posted 8 months ago

Question
Solve for $x$ . $\newline$ $7 = 9^x$ $\newline$ Round your answer to the nearest thousandth.
Get tutor help
Posted 4 months ago

Question
Solve. Round your answer to the nearest thousandth. $\newline$ $7 = e^x$ $\newline$ $x =$ ____
Get tutor help
Posted 7 months ago

Question
Solve. Simplify your answer. $\newline$ $\log u = 1$ $\newline$ $u =$ ____
Get tutor help
Posted 9 months ago

Question
Solve. Simplify your answer. $\newline$ $7\log x = 7$ $\newline$ $x =$ ____
Get tutor help
Posted 7 months ago

Question
How does $g(t) = 3^t$ change over the interval from $t = 3$ to $t = 4$ ? $\newline$ Choices: $\newline$ (A) $g(t)$ decreases by $3$ $\newline$ (B) $g(t)$ increases by $3$ $\newline$ (C) $g(t)$ decreases by $3\%$ $\newline$ (D) $g(t)$ increases by $200\%$
Get tutor help
Posted 4 months ago

Question
A function $f(x)$ increases by $6$ over every unit interval in $x$ and $f(0) = 0$ . $\newline$ Which could be a function rule for $f(x)$ ? $\newline$ Choices: $\newline$ (A) $f(x) = 6x$ $\newline$ (B) $f(x) = 6^x$ $\newline$ (C) $f(x) = \frac{1}{6^x}$ $\newline$ (D) $f(x) = \frac{x}{6}$
Get tutor help
Posted 4 months ago

View all (200)

If $-y+x+y^{2}+5 y^{3}+3 x^{2}=0$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

Full solution

More problems from Evaluate exponential functions

Related topics

If −y+x+y2+5y3+3x2=0 -y+x+y^{2}+5 y^{3}+3 x^{2}=0 −y+x+y2+5y3+3x2=0 then find dydx \frac{d y}{d x} dxdy​ in terms of x x x and y y y.\newlineAnswer: dydx= \frac{d y}{d x}= dxdy​=

If $-y+x+y^{2}+5 y^{3}+3 x^{2}=0$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$