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Using implicit differentiation, find (dy)/(dx). -7cos(3x)sin(3y)=4x-1

Using implicit differentiation, find dydx \frac{d y}{d x} .\newline−7cos⁡(3x)sin⁡(3y)=4x−1 -7 \cos (3 x) \sin (3 y)=4 x-1

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Q. Using implicit differentiation, find dydx \frac{d y}{d x} .\newline−7cos⁡(3x)sin⁡(3y)=4x−1 -7 \cos (3 x) \sin (3 y)=4 x-1
  1. Given Equation: We are given the equation −7cos⁡(3x)sin⁡(3y)=4x−1-7\cos(3x)\sin(3y)=4x-1. To find dydx\frac{dy}{dx}, we will differentiate both sides of the equation with respect to xx, using the chain rule for the functions of yy, since yy is a function of xx.
  2. Differentiate Left Side: Differentiate the left side of the equation with respect to xx. The derivative of −7cos⁡(3x)sin⁡(3y)-7\cos(3x)\sin(3y) with respect to xx is −7[−sin⁡(3x)(3)(sin⁡(3y))+cos⁡(3x)(3)(cos⁡(3y))(dydx)]-7[-\sin(3x)(3)(\sin(3y)) + \cos(3x)(3)(\cos(3y))(\frac{dy}{dx})] by using the product rule and chain rule.
  3. Differentiate Right Side: Differentiate the right side of the equation with respect to xx. The derivative of 4x−14x-1 with respect to xx is 44, since the derivative of a constant is 00.
  4. Simplify Left Side: Now we have the equation −7[−sin⁡(3x)(3)(sin⁡(3y))+cos⁡(3x)(3)(cos⁡(3y))(dydx)]=4-7[-\sin(3x)(3)(\sin(3y)) + \cos(3x)(3)(\cos(3y))\left(\frac{dy}{dx}\right)] = 4. Simplify the left side by distributing the −7-7 and the 33 inside the brackets.
  5. Isolate dydx\frac{dy}{dx} Term: After simplifying, we get 21sin⁡(3x)sin⁡(3y)−21cos⁡(3x)cos⁡(3y)dydx=421\sin(3x)\sin(3y) - 21\cos(3x)\cos(3y)\frac{dy}{dx} = 4.
  6. Move Term to Other Side: We want to solve for dydx\frac{dy}{dx}, so we need to isolate the term containing dydx\frac{dy}{dx}. Move the term without dydx\frac{dy}{dx} to the other side of the equation by adding 21sin⁡(3x)sin⁡(3y)21\sin(3x)\sin(3y) to both sides.
  7. Divide by Constant: We now have −21cos⁡(3x)cos⁡(3y)(dydx)=4+21sin⁡(3x)sin⁡(3y)-21\cos(3x)\cos(3y)\left(\frac{dy}{dx}\right) = 4 + 21\sin(3x)\sin(3y).
  8. Final Expression: Divide both sides of the equation by −21cos⁡(3x)cos⁡(3y)-21\cos(3x)\cos(3y) to solve for (dydx)(\frac{dy}{dx}).
  9. Final Expression: Divide both sides of the equation by −21cos⁡(3x)cos⁡(3y)-21\cos(3x)\cos(3y) to solve for dydx\frac{dy}{dx}.The final expression for dydx\frac{dy}{dx} is dydx=4+21sin⁡(3x)sin⁡(3y)−21cos⁡(3x)cos⁡(3y)\frac{dy}{dx} = \frac{4 + 21\sin(3x)\sin(3y)}{-21\cos(3x)\cos(3y)}.

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