dy/dx of xy^2=tan(2x+4y)
i got the answer up to
2xy dy/dx+y^2=sec^2(2x+4y)(2+4 dy/dx) but i can figure out the rest :(
Your derivative so fare is correct, now
expand your right side to ..
2sec^2(2x+4y) + 4 dy/dx sec^2(2x+4y)
bring both terms containing dy/dx to one side, the other two terms to the other side.
Factor out dy/dx, then isolated dy/dx by dividing by the remaining factor.
I see it as
dy/dx = (2sec^2(2x+4y) - y^2)/(2xy - 4sec^2(2x+4y))
THanks Reiny :)
To find dy/dx of xy^2 = tan(2x + 4y), we need to use implicit differentiation. Here's how you can do it step by step:
Step 1: Write down the given equation:
xy^2 = tan(2x + 4y)
Step 2: Differentiate both sides of the equation with respect to x:
(d/dx)(xy^2) = (d/dx)(tan(2x + 4y))
Step 3: Apply the product rule on the left side of the equation:
(d/dx)(x)(y^2) + (x)(d/dx)(y^2) = (d/dx)(tan(2x + 4y))
Step 4: Simplify the left side using the chain rule:
y^2 + 2xy(dy/dx) = (d/dx)(tan(2x + 4y))
Step 5: Now, let's focus on finding (d/dx)(tan(2x + 4y)). To differentiate tan(2x + 4y), we need to apply the chain rule. The derivative of tan(u) with respect to u is sec^2(u), multiplied by the derivative of the inside function. In this case, the inside function is 2x + 4y.
So, (d/dx)(tan(2x + 4y)) = sec^2(2x + 4y) * (d/dx)(2x + 4y)
Step 6: Simplify the right side using the chain rule:
(d/dx)(2x + 4y) = 2 + 4(dy/dx)
Step 7: Replace the right side of the equation:
y^2 + 2xy(dy/dx) = sec^2(2x + 4y) * (2 + 4(dy/dx))
Step 8: Distribute sec^2(2x + 4y) to both terms on the right side:
y^2 + 2xy(dy/dx) = 2sec^2(2x + 4y) + 4(dy/dx)sec^2(2x + 4y)
Step 9: Rearrange the equation to isolate dy/dx on one side:
2xy(dy/dx) - 4(dy/dx)sec^2(2x + 4y) = 2sec^2(2x + 4y) - y^2
Step 10: Factor out (dy/dx) on the left side:
(dy/dx)(2xy - 4sec^2(2x + 4y)) = 2sec^2(2x + 4y) - y^2
Step 11: Solve for dy/dx by dividing both sides by (2xy - 4sec^2(2x + 4y)):
dy/dx = (2sec^2(2x + 4y) - y^2) / (2xy - 4sec^2(2x + 4y))
And that's the derivative, dy/dx, of the given equation xy^2 = tan(2x + 4y).