Differentiation of Implicit Function
The given equation is y^2 = 9x^2 - 4x. To find dy/dx, differentiate both sides of the equation with respect to x, applying the chain rule to the term involving y because y is a function of x.
Differentiating y^2 with respect to x yields:
2y * dy/dx (using the chain rule because y is a function of x)
Differentiating 9x^2 - 4x with respect to x yields:
18x - 4
Now, equate these two results:
2y * dy/dx = 18x - 4
To solve for dy/dx, we divide both sides by 2y:
dy/dx = (18x - 4) / (2y)
To simplify the expression, we divide each term in the numerator by 2:
dy/dx = (9x - 2) / y
Now we have dy/dx. To find d^2y/dx^2, we need to differentiate dy/dx with respect to x using the quotient rule. Let's call u = 9x - 2 and v = y. The quotient rule states that if you have a function g(x) = u/v, its derivative g'(x) is given by:
g'(x) = (u'v - uv') / v^2
Differentiating u = 9x - 2 with respect to x gives us u' = 9.
Differentiating v = y with respect to x gives us v' = dy/dx.
So applying the quotient rule gives us:
d^2y/dx^2 = (9 * y - (9x - 2) * dy/dx) / y^2
Since we've found dy/dx previously as (9x - 2) / y, we substitute that in for dy/dx in the above expression:
d^2y/dx^2 = (9 * y - (9x - 2) * ((9x - 2) / y)) / y^2
Let's simplify this expression:
d^2y/dx^2 = (9 * y^2 - (9x - 2) * (9x - 2)) / y^3
At this point, it would be best to expand the numerator and then simplify the expression for the second derivative.
d^2y/dx^2 = (9y^2 - (81x^2 - 18x*2 + 2^2)) / y^3
d^2y/dx^2 = (9y^2 - (81x^2 - 36x + 4)) / y^3
d^2y/dx^2 = (9y^2 - 81x^2 + 36x - 4) / y^3
Now we have both the first and second derivatives:
dy/dx = (9x - 2) / y
d^2y/dx^2 = (9y^2 - 81x^2 + 36x - 4) / y^3
