Using implicit differentiation, it is found that:
[tex]\frac{dy}{dx} = \frac{5y^2\sin{5x}+6y\ln{y}}{y\cos{5x}+2y^2+6x}[/tex]
The function is:
[tex]y\cos{5x} + y^2 = 6x\ln{y}[/tex]
In implicit differentiation:
- We apply the standard derivative rules.
- The numerator is the variable we derived.
- The denominator is fixed.
Applying implicit differentiation:
[tex]\cos{5x}\frac{dy}{dx} - 5y\sin{5x}\frac{dx}{dx} + 2y\frac{dy}{dx} = 6\ln{y}\frac{dx}{dx} + \frac{6x}{y}\frac{dy}{dx}[/tex]
[tex]\cos{5x}\frac{dy}{dx} + 2y\frac{dy}{dx} + \frac{6x}{y}\frac{dy}{dx} = 5y\sin{5x} + 6\ln{y}[/tex]
[tex]\frac{dy}{dx}(\cos{5x} + 2y + \frac{6x}{y}) = 5y\sin{5x} + 6\ln{y}[/tex]
[tex]\frac{dy}{dx}\left(\frac{y\cos{5x}+2y^2+6x}{y}\right) = 5y\sin{5x} + 6\ln{y}[/tex]
[tex]\frac{dy}{dx} = \frac{5y^2\sin{5x}+6y\ln{y}}{y\cos{5x}+2y^2+6x}[/tex]
A similar problem is given at https://brainly.com/question/9543179
