sinx+x^2y=1 use implicit differentiation to find the second derivative. For the first derivative I got (cos(x)+2xy)/x^2. I have a feeling that is wrong, can someone please help and show steps on how to do this problem. (studying for a midterm).
cos x dx + x^2 dy + 2 x y dx = 0
x^2 dy = - (cos x +2 xy)dx
dy/dx = - (cos x + 2 x y)/x^2
To find the second derivative using implicit differentiation, let's start by differentiating both sides of the given equation with respect to x.
The given equation is: sin(x) + x^2y = 1
Taking the derivative of both sides with respect to x:
d/dx (sin(x) + x^2y) = d/dx (1)
Applying the derivative rules:
cos(x) + d/dx (x^2y) = 0
To find d/dx (x^2y), we will use the product rule:
d/dx (x^2y) = (d/dx (x^2))y + x^2(d/dx (y))
Differentiating each term of the product rule:
d/dx (x^2) = 2x
d/dx (y) = dy/dx (since y is a function of x)
Now, let's substitute these results back into the equation:
cos(x) + 2xy + x^2(dy/dx) = 0
To find the first derivative, solve for dy/dx:
dy/dx = -(cos(x) + 2xy) / x^2
Now, to find the second derivative, we need to differentiate the first derivative with respect to x. Applying the quotient rule:
d^2y/dx^2 = (x^2(2x(dy/dx) - (cos(x) + 2xy)(2)) - (-(cos(x) + 2xy)(2x)) / (x^4)
Simplifying:
d^2y/dx^2 = (2x^3(dy/dx) - 2x(cos(x) + 2xy) + 2x(cos(x) + 2xy)) / x^4
Simplifying further:
d^2y/dx^2 = (2x^3(dy/dx)) / x^4
Finally, substitute the first derivative into the equation:
d^2y/dx^2 = (2x^3(-(cos(x) + 2xy) / x^2)) / x^4
Simplifying:
d^2y/dx^2 = (-2x^2(cos(x) + 2xy)) / x^4
The second derivative is given by:
d^2y/dx^2 = -2(cos(x) + 2xy) / x^2