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

If $5 y+y^{3}-y^{2}-x^{3}=0$ 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

5 y+y^{3}-y^{2}-x^{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 $5y + y^3 - y^2 - x^3 = 0$ 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 $y$ , we will apply the chain rule, which states that the derivative of a function of a function is the derivative of the outer function times the derivative of the inner function. In this case, the ＂inner function＂ is $y$ , which is a function of $x$ , so we multiply by $\frac{dy}{dx}$ after differentiating with respect to $y$ .
Differentiating $5y$ : Differentiating $5y$ with respect to $x$ gives us $5\frac{dy}{dx}$ , since the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$ .
Differentiating $y^3$ : Differentiating $y^3$ with respect to $x$ gives us $3y^2\frac{dy}{dx}$ , by applying the chain rule (the derivative of $y^3$ with respect to $y$ is $3y^2$ , and then we multiply by $\frac{dy}{dx}$ ).
Differentiating $-y^2$ : Differentiating $-y^2$ with respect to $x$ gives us $-2y\frac{dy}{dx}$ , by applying the chain rule (the derivative of $y^2$ with respect to $y$ is $2y$ , and then we multiply by $\frac{dy}{dx}$ .
Differentiating $-x^3$ : Differentiating $-x^3$ with respect to $x$ gives us $-3x^2$ , since $x$ is the variable we are differentiating with respect to.
Combining Differentiated Terms: Now we combine all the differentiated terms to rewrite the equation: $5\left(\frac{dy}{dx}\right) + 3y^2\left(\frac{dy}{dx}\right) - 2y\left(\frac{dy}{dx}\right) - 3x^2 = 0$ .
Factoring out $\frac{dy}{dx}$ : We can factor out $\frac{dy}{dx}$ from the terms that contain it: $\left(\frac{dy}{dx}\right)(5 + 3y^2 - 2y) - 3x^2 = 0$ .
Isolating $\frac{dy}{dx}$ : Now we isolate $\frac{dy}{dx}$ by adding $3x^2$ to both sides and then dividing by $(5 + 3y^2 - 2y)$ : $\frac{dy}{dx} = \frac{3x^2}{5 + 3y^2 - 2y}$ .

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