Suppose
e^y=sin(x+cosy)
Find dy/dx using implicit differentiation
To find dy/dx using implicit differentiation, follow these steps:
1. Start by differentiating both sides of the equation with respect to x.
- Differentiate e^y with respect to x using the chain rule: (d/dx)e^y = (d/dy)e^y * (dy/dx) = e^y * (dy/dx)
- Differentiate sin(x + cosy) with respect to x using the chain rule: (d/dx)sin(x + cosy) = cos(x + cosy) * (d/dx)(x + cosy) = cos(x + cosy) * (1 + (d/dx)cosy)
2. After differentiating, rearrange the equation to isolate (dy/dx) on one side:
e^y * (dy/dx) = cos(x + cosy) * (1 + (d/dx)cosy)
3. Solve for (dy/dx) by dividing both sides by e^y and simplifying:
(dy/dx) = (cos(x + cosy) * (1 + (d/dx)cosy)) / e^y
Note: To find (d/dx)cosy, we need to use the chain rule again, treating cosy as a composite function. The derivative of cosy with respect to x can be found by taking the derivative of cosy with respect to y and then multiplying it by (dy/dx).
Overall, the process involves differentiating each term with respect to x, applying the chain rule as needed, and isolating (dy/dx) in the final equation.