Question

Help me find the derivative of `sinx+x^2y=10`?

Answer 1

`(2sin x - x cos x-20)/x^3`

Explanation:

Differentiating both sides of

`sinx+x^2y=10`

with respect to `x` leads to

`cos x +2xy+x^2dy/dx=0`

so that

`dy/dx = -(cos x+2xy)/(x^2) = -1/x^2 cos x -2/x y`

We could solve the original expression for `y` to get

`y = (10-sin x)/x^2`

and substituting this will give us

`dy/dx = -1/x^2 cos x -2/x (10-sin x)/x^2`
`qquad qquad = (2sin x - x cos x-20)/x^3`

Note that you could have also obtained this by using the explicit expression for `y` and then using the quotient rule.