If 3 x2+ 5 x + xy = 3 and y( 3 ) = −13, find y′( 3 ) by implicit differentiation.
To find the derivative of y with respect to x, we need to use implicit differentiation.
First, differentiate both sides of the equation with respect to x:
d/dx (3x^2 + 5x + xy) = d/dx (3)
Now, let's differentiate each term on the left side:
d/dx (3x^2) + d/dx (5x) + d/dx (xy) = 0
The derivative of 3x^2 with respect to x is 6x:
6x + 5 + d/dx (xy) = 0
Now, we need to apply the product rule to differentiate xy. The product rule states that the derivative of the product of two functions u(x) * v(x) is given by:
d/dx (u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = x and v(x) = y. So, applying the product rule:
x * d/dx(y) + y * d/dx(x) = 6x + 5
Since we are looking for y'(3), we can substitute x = 3 into the equation:
3 * y' + y * 1 = 6(3) + 5
Now, we know that y(3) = -13, so we can substitute y = -13:
3 * y' + (-13) = 18 + 5
3 * y' - 13 = 23
Finally, we can solve for y':
3 * y' = 23 + 13
3 * y' = 36
y' = 12
Therefore, y'(3) = 12.