arcsinxy = 2/3 arctan4x find dy/dx at the point (1/4, 2)
using implicit differentiation,
(xy'+y)/√(1+x^2y^2) = 8/3 * 1/(1+16x^2)
a little rearranging of terms yields
y' = 8√(1+x^2y^2) / (3x(1+16x^2)) - y/x
oops that's 1-x^2y^2 throughout
when I plug in the ordered pair (.25, 2)I get a zero in the denominator, was I doing that wrong?
√(1-x^2y^2) = 0, all right, but it's not in the denominator.
So, all you end up with is -y/x = -16
oops: -2/.25 = -8
Rats! I keep being inconsistent in the squaring! √(1-x^2y^2) is not zero.
at (1/4,2)
y' = 8√(1-1/4)/(3*1/4*2) - 2/.25
= (8√6)/3 - 8
judging from my other posts, you'd better double-check this one too!
To find dy/dx, we can differentiate both sides of the equation arcsin(xy) = (2/3)arctan(4x) with respect to x.
Using the chain rule, we have:
(d/dx)arcsin(xy) = (d/dx)((2/3)arctan(4x))
To differentiate the left side, we need to use the chain rule again. Let's declare another variable u = xy.
So, (d/dx)arcsin(xy) = (d/dx)arcsin(u)
Applying the chain rule, we get:
(d/dx)arcsin(u) = (d/du)arcsin(u) * (du/dx)
The derivative of arcsin(u) with respect to u is 1/sqrt(1-u^2), and the derivative of u = xy with respect to x is y.
So, (d/dx)arcsin(u) = (1/sqrt(1-u^2)) * y
Substituting back u = xy, we have:
(d/dx)arcsin(xy) = (1/sqrt(1-(xy)^2)) * y
Now focusing on the right side of the equation, we have:
(d/dx)((2/3)arctan(4x))
Using the chain rule, the derivative of arctan(4x) with respect to x is 4/(1+(4x)^2). So, we have:
(d/dx)((2/3)arctan(4x)) = (2/3) * (4/(1+(4x)^2))
Simplifying the expression, we get:
(d/dx)((2/3)arctan(4x)) = 8/(3+12x^2)
Now we can equate the derivatives of both sides of the equation and solve for dy/dx:
(1/sqrt(1-(xy)^2)) * y = 8/(3+12x^2)
To find dy/dx at the point (1/4, 2), we substitute x = 1/4 and y = 2 into the equation:
(1/sqrt(1-(1/4)(2)^2)) * 2 = 8/(3+12(1/4)^2)
Simplifying further:
(1/sqrt(1-1/8)) * 2 = 8/(3+3/16)
(1/sqrt(7/8)) * 2 = 8/(51/16)
(1/sqrt(7/8)) * 2 = 128/51
Multiplying both sides by sqrt(7/8):
(2/sqrt(7/8))^2 = (128/51)^2
(4/√7)^2 = (128/51)^2
16/7 = 16384/2601
Cross-multiplying:
16 * 2601 = 16384 * 7
41616 = 114688
Since the equation is not true, there might be a mistake in the calculations or in the equation itself. Double-check your equation and calculations to ensure accuracy.