Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

If 5=xy33y33x 5=-x y^{3}-3 y^{3}-3 x then find dydx \frac{d y}{d x} in terms of x x and y y .\newlineAnswer: dydx= \frac{d y}{d x}=

Full solution

Q. If 5=xy33y33x 5=-x y^{3}-3 y^{3}-3 x then find dydx \frac{d y}{d x} in terms of x x and y y .\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Given Equation: We are given the equation 5=xy33y33x5 = -xy^{3} - 3y^{3} - 3x, and we need to find the derivative of yy with respect to xx, which is dydx\frac{dy}{dx}. To do this, we will use implicit differentiation, treating yy as a function of xx.
  2. Implicit Differentiation: Differentiate both sides of the equation with respect to xx. Remember that when differentiating terms with yy, we treat yy as a function of xx and use the chain rule to include dydx\frac{dy}{dx}.ddx[5]=ddx[xy3]+ddx[3y3]+ddx[3x]\frac{d}{dx}[5] = \frac{d}{dx}[-x y^{3}] + \frac{d}{dx}[-3 y^{3}] + \frac{d}{dx}[-3x]
  3. Product Rule: The derivative of a constant is 00, so ddx[5]=0\frac{d}{dx}[5] = 0. For ddx[xy3]\frac{d}{dx}[-xy^{3}], we use the product rule: ddx[uv]=uv+uv\frac{d}{dx}[uv] = u'v + uv', where u=xu = x and v=y3v = y^{3}. So, ddx[xy3]=1×y3+x×3y2×dydx\frac{d}{dx}[-xy^{3}] = -1 \times y^{3} + -x \times 3y^{2} \times \frac{dy}{dx}. For ddx[3y3]\frac{d}{dx}[-3y^{3}], we simply multiply the derivative of y3y^{3} by 3-3, which gives us ddx[5]=0\frac{d}{dx}[5] = 000. For ddx[5]=0\frac{d}{dx}[5] = 011, the derivative is simply 3-3.
  4. Combining Differentiated Terms: Now we combine all the differentiated terms:\newline0=y33xy2dydx9y2dydx3.0 = -y^{3} - 3x \cdot y^{2} \cdot \frac{dy}{dx} - 9y^{2} \cdot \frac{dy}{dx} - 3.
  5. Isolating Terms: We need to solve for dydx\frac{dy}{dx}. First, we isolate the terms that contain dydx\frac{dy}{dx} on one side of the equation:\newline3xy2dydx+9y2dydx=y333x \cdot y^{2} \cdot \frac{dy}{dx} + 9y^{2} \cdot \frac{dy}{dx} = y^{3} - 3.
  6. Factoring Out: Factor out (dydx)(\frac{dy}{dx}) from the left side of the equation:\newlinedydx(3xy2+9y2)=y33\frac{dy}{dx} \cdot (3x \cdot y^{2} + 9y^{2}) = y^{3} - 3.
  7. Dividing Both Sides: Now, divide both sides by (3xy2+9y2)(3x \cdot y^{2} + 9y^{2}) to solve for dydx\frac{dy}{dx}:dydx=y333xy2+9y2\frac{dy}{dx} = \frac{y^{3} - 3}{3x \cdot y^{2} + 9y^{2}}.
  8. Simplifying the Denominator: We can simplify the denominator by factoring out 3y23y^{2}:dydx=y333y2(x+3)\frac{dy}{dx} = \frac{y^{3} - 3}{3y^{2} \cdot (x + 3)}.

More problems from Evaluate exponential functions