Find dy/dx 2x+3y=sinx
To find dy/dx, we need to take the derivative of both sides of the equation with respect to x.
Let's start by differentiating the left side of the equation, 2x + 3y, with respect to x using the sum rule of differentiation.
The sum rule states that if we have two functions u(x) and v(x), then the derivative of their sum is equal to the sum of their derivatives.
For the first term, 2x, the derivative with respect to x is simply 2 (since the derivative of x with respect to x is 1).
For the second term, 3y, we need to use the chain rule because y is a function of x. The chain rule states that if we have a function y = f(u), and u = g(x), then the derivative of y with respect to x is equal to the derivative of f(u) with respect to u, multiplied by the derivative of g(x) with respect to x.
In this case, u = y and f(u) = 3u. Therefore, the derivative of 3y with respect to x is 3 times the derivative of y with respect to x, which is 3(dy/dx).
Now, let's differentiate the right side of the equation, sin(x), with respect to x.
The derivative of sin(x) with respect to x is cos(x).
Now we can equate the derivatives of both sides:
2 + 3(dy/dx) = cos(x)
To isolate dy/dx, we subtract 2 from both sides:
3(dy/dx) = cos(x) - 2
Finally, to solve for dy/dx, we divide both sides by 3:
dy/dx = (cos(x) - 2) / 3
Therefore, the derivative of 2x + 3y = sin(x) with respect to x is (cos(x) - 2) / 3.