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Calculate the derivative of yy with respect to xx. Express derivative in terms of xx and yy.\newlinee2xy=sin⁡(y7)e^{2xy} = \sin(y^{7})\newline(Express numbers in exact form. Use symbolic notation and fractions where needed.)

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Q. Calculate the derivative of yy with respect to xx. Express derivative in terms of xx and yy.\newlinee2xy=sin⁡(y7)e^{2xy} = \sin(y^{7})\newline(Express numbers in exact form. Use symbolic notation and fractions where needed.)
  1. Apply Implicit Differentiation: First, we need to apply implicit differentiation to both sides of the equation with respect to xx. The left side of the equation involves the exponential function e2xye^{2xy}, and the right side involves the sine function sin⁡(y7)\sin(y^{7}). We will need to use the chain rule for both sides.
  2. Differentiate Left Side: Differentiate the left side e2xye^{2xy} with respect to xx. The derivative of eue^{u} with respect to xx is eu⋅(dudx)e^{u} \cdot \left(\frac{du}{dx}\right), where uu is a function of xx. Here, u=2xyu = 2xy, so we need to find the derivative of 2xy2xy with respect to xx, which is xx00 because we treat xx11 as a function of xx.
  3. Differentiate Right Side: Differentiate the right side sin⁡(y7)\sin(y^{7}) with respect to xx. The derivative of sin⁡(v)\sin(v) with respect to xx is cos⁡(v)⋅(dvdx)\cos(v) \cdot (\frac{dv}{dx}), where vv is a function of xx. Here, v=y7v = y^{7}, so we need to find the derivative of y7y^{7} with respect to xx, which is xx00.
  4. Write Differentiated Form: Now we write the differentiated form of the equation: e2xy⋅(2y+2xdydx)=cos⁡(y7)⋅(7y6⋅dydx)e^{2xy} \cdot (2y + 2x\frac{dy}{dx}) = \cos(y^{7}) \cdot (7y^{6} \cdot \frac{dy}{dx}).
  5. Solve for dydx\frac{dy}{dx}: We need to solve for dydx\frac{dy}{dx}, the derivative of yy with respect to xx. To do this, we'll isolate terms involving dydx\frac{dy}{dx} on one side of the equation. Let's move all terms not involving dydx\frac{dy}{dx} to the other side.
  6. Rearrange Equation: Rearrange the equation to isolate terms with dydx\frac{dy}{dx}: e2xy⋅2y=cos⁡(y7)⋅(7y6⋅dydx)−e2xy⋅2x⋅dydxe^{2xy} \cdot 2y = \cos(y^{7}) \cdot (7y^{6} \cdot \frac{dy}{dx}) - e^{2xy} \cdot 2x \cdot \frac{dy}{dx}.
  7. Factor out dydx\frac{dy}{dx}: Factor out dydx\frac{dy}{dx} from the right side of the equation: e2xy⋅2y=(dydx)⋅(7y6⋅cos⁡(y7)−2x⋅e2xy)e^{2xy} \cdot 2y = \left(\frac{dy}{dx}\right) \cdot \left(7y^{6} \cdot \cos(y^{7}) - 2x \cdot e^{2xy}\right).
  8. Divide to Solve for dydx\frac{dy}{dx}: Divide both sides by (7y6⋅cos⁡(y7)−2x⋅e2xy)(7y^{6} \cdot \cos(y^{7}) - 2x \cdot e^{2xy}) to solve for dydx\frac{dy}{dx}: dydx=e2xy⋅2y7y6⋅cos⁡(y7)−2x⋅e2xy\frac{dy}{dx} = \frac{e^{2xy} \cdot 2y}{7y^{6} \cdot \cos(y^{7}) - 2x \cdot e^{2xy}}.

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