Calculate dy/dx, (x^3)(y^7) + 16xy^6=0
My ans is (y(-16-3x^2y))/ x(96+7x^2y) but at wolfram is y^6(3x^2y+16). Btw i use product rule to dy/dx.
Need someone to recheck my ans.
3x^2y^7 + 7x^3y^6y' + 16y^6 + 96xy^5 y' = 0
y'(7x^3y^6+96xy^5) = -(3x^2y^7 + 16y^6)
y' =
-y^6 (3x^2 y + 16)
---------------------
xy^5 (7x^2 y + 96)
y' =
-y(3x^2y+16)
-----------------
x(7x^2 y + 96)
you are correct, and Wolfram agrees for me. If the answer you got was different, you must have mistyped it into Wolfram
Thanks guy
To find the derivative of the given equation, we can use the product rule. The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
d/dx(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
Let's find the derivative step by step:
Given equation: (x^3)(y^7) + 16xy^6 = 0
Taking the derivative of both sides with respect to x:
d/dx[(x^3)(y^7)] + d/dx[16xy^6] = 0
Now let's find the derivative of each term separately.
For the first term, we have u(x) = x^3 and v(x) = y^7:
u'(x) = d/dx(x^3) = 3x^2
v(x) = y^7 (treat y as a constant because we're differentiating with respect to x)
Using the product rule: [u'(x) * v(x)] + [u(x) * v'(x)]
= (3x^2)(y^7) + (x^3)(d/dx(y^7))
To find d/dx(y^7), we need to use the chain rule since y is also a function of x. The chain rule states that for a composite function y(u(x)), the derivative with respect to x is given by:
d/dx[y(u(x))] = (dy/du) * (du/dx)
In this case, u(x) = y and y^7 is our composite function.
Using the chain rule:
d/dx(y^7) = (dy/du) * (du/dx) = (7y^6) * dy/dx
So the first term becomes: (3x^2)(y^7) + (x^3)(7y^6)(dy/dx)
For the second term, u(x) = 16x and v(x) = y^6 (treat y as a constant again):
u'(x) = d/dx(16x) = 16
v(x) = y^6
Using the product rule: [u'(x) * v(x)] + [u(x) * v'(x)]
= (16)(y^6) + (16x)(d/dx(y^6))
Now, applying the chain rule to find d/dx(y^6):
d/dx(y^6) = (dy/du) * (du/dx) = (6y^5) * dy/dx
So the second term becomes: (16)(y^6) + (16x)(6y^5)(dy/dx)
Now, combining the two terms: (3x^2)(y^7) + (x^3)(7y^6)(dy/dx) + (16)(y^6) + (16x)(6y^5)(dy/dx)
We can group together the terms that contain dy/dx: (7x^3y^6)(dy/dx) + (16x)(6y^5)(dy/dx)
Factoring out dy/dx, we get: dy/dx[(7x^3y^6) + (16x)(6y^5)]
Finally, simplifying the expression:
dy/dx = -[(7x^3y^6) + (16x)(6y^5)] / [(x^3)(y^7) + 16xy^6]
To compare your solution with Wolfram Alpha, you should perform the simplification steps further to see if the two expressions match.