If 2x2+5x+xy=3 and y(3)=−10, find y′(3) by implicit differentiation.

Question

If 2x2+5x+xy=32x2+5x+xy=3 and y(3)=−10y(3)=−10, find y′(3)y′(3) by implicit differentiation.

Answer 1

Assuming the problem is:

Take the derivative of each term in the equation:

The derivative of 2x² (using the power rule) is 4x

The derivative of 5x is 5

The derivative of xy (since it is the product of x and y) is done via the product rule which is (u•v)' = u'v + uv'. So the derivative of xy is x'y + xy' = 1•y + xy' = y + xy'

The derivative of 3 is zero.

So, putting those together, you get:

4x + 5 + y + xy' = 0

xy' = -4x - 5 - y

y' = (-4x - 5 - y)/x

Since y(3) = −10 we can say (that for y'(3)) x = 3 and y = -10 so:

y'(3) = (-4•3 - 5 - -10)/3 = (-12 - 5 + 10)/3 = -7/3