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

Differentiation Process: To find the derivative of y with respect to x, we need to differentiate both sides of the equation with respect to x. The equation is y = 5y^3 + 2x^3.Derivative of y: Differentiate the left side of the equation with respect to x. The derivative of y with respect to x is simply dy/dx.Derivative of Right Side: Differentiate the right side of the equation with respect to x. The term 5y^3 is treated as a function of x, so we apply the chain rule. The derivative of 5y^3 with respect to x is 15y^2 * (dy/dx). The derivative of 2x^3 with respect to x is 6x^2.Combining Derivatives: Combine the derivatives to form the equation: dy/dx = 15y^2 * (dy/dx) + 6x^2.Isolating dy/dx: To solve for dy/dx, we need to move all terms involving dy/dx to one side of the equation. Subtract 15y^2 * (dy/dx) from both sides to get: dy/dx - 15y^2 * (dy/dx) = 6x^2.Factoring out dy/dx: Factor out dy/dx on the left side of the equation to get: dy/dx * (1 - 15y^2) = 6x^2.Final Derivative Solution: Divide both sides of the equation by (1 - 15y^2) to solve for dy/dx. This gives us: dy/dx = 6x^2 / (1 - 15y^2).

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

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

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Math Problems

Algebra 1

Evaluate an exponential function

Full solution

Q. If

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

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Differentiation Process: To find the derivative of $y$ with respect to $x$ , we need to differentiate both sides of the equation with respect to $x$ . The equation is $y = 5y^3 + 2x^3$ .
Derivative of $y$ : Differentiate the left side of the equation with respect to $x$ . The derivative of $y$ with respect to $x$ is simply $\frac{dy}{dx}$ .
Derivative of Right Side: Differentiate the right side of the equation with respect to $x$ . The term $5y^3$ is treated as a function of $x$ , so we apply the chain rule. The derivative of $5y^3$ with respect to $x$ is $15y^2 \cdot \frac{dy}{dx}$ . The derivative of $2x^3$ with respect to $x$ is $6x^2$ .
Combining Derivatives: Combine the derivatives to form the equation: $\frac{dy}{dx} = 15y^2 \cdot \left(\frac{dy}{dx}\right) + 6x^2$ .
Isolating $\frac{dy}{dx}$ : To solve for $\frac{dy}{dx}$ , we need to move all terms involving $\frac{dy}{dx}$ to one side of the equation. Subtract $15y^2 \times \frac{dy}{dx}$ from both sides to get: $\frac{dy}{dx} - 15y^2 \times \frac{dy}{dx} = 6x^2$ .
Factoring out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ on the left side of the equation to get: $\frac{dy}{dx} \times (1 - 15y^2) = 6x^2$ .
Final Derivative Solution: Divide both sides of the equation by $(1 - 15y^2)$ to solve for $\frac{dy}{dx}$ . This gives us: $\frac{dy}{dx} = \frac{6x^2}{(1 - 15y^2)}$ .