find dy/dx 3x^2y^2-5sin(xy) +7x =25y+2

Question

Find dy/dx of the function

Answer 1

Use implicit differentiation:

3x²y² - 5sin(xy) + 7x = 25y + 2

Take the derivative of each element with respect to "x":

To find the derivative of 3x²y², use the product rule (u•v)' = u'v + uv' where u = 3x² meaning u' = 6x and v = y² meaning v' = 2y•y' (must use the chain rule). Putting these together, the derivative of 3x²y² is (6x)(y²) + (3x²)(2y•y') = 6xy² + 6x²y•y'

To find the derivative of -5sin(xy) we must use the chain rule. It would be -5cos(xy)•(xy)' but to get the derivative of (xy)' we must use the product rule. (xy)' = x'y + xy' = y + xy' meaning the derivative of -5sin(xy) is -5cos(xy)•(y + xy')

The derivative of 7x is 7

The derivative of 25y is 25y'

The derivative of 2 is zero.

Putting these all together we get:

6xy² + 6x²y•y' - 5cos(xy)•(y + xy') + 7 = 25y'

Now, solve for y':

6xy² + 6x²y•y' - 5cos(xy)•(y + xy') + 7 = 25y'

6xy² + 6x²y•y' - 5ycos(xy) - 5xy'cos(xy) + 7 = 25y'

6x²y•y' - 5xy'cos(xy) - 25y' = 5ycos(xy) - 6xy² - 7

y'(6x²y - 5xcos(xy) - 25) = 5ycos(xy) - 6xy² - 7

y' = (5ycos(xy) - 6xy² - 7)/(6x²y - 5xcos(xy) - 25)