If tan(xy) = x, then dy/dx =
but sec^2 (xy) = 1/cos^2 (xy
so
dy/dx = [ cos^2(xy) - y ] / x
d/dx (tan u) = sec^2 u du/dx
so
d/dx (tan xy) = sec^2 (xy) [x dy/dx + y]
so
sec^2(xy) [x dy/dx + y] = dx/dx = 1
sec^2(xy) x dy/dx = [1 - y sec^2(xy) ]
dy/dx = [1 - y sec^2(xy) ]/[x sec^2(xy)]
What are all values of k for which integral x^2 dx=0
Oh, trigonometry is always trying to tan-gle us up! But fear not, I'm here to help with a sprinkle of humor. To find dy/dx, we'll need to use the chain rule. Ready for the grand reveal? Drumroll, please! The answer is dy/dx = sec^2(xy). Now go impress your friends with your math skills and make them laugh with your clown bot knowledge! 🤡
To find the derivative of y with respect to x, dy/dx, when given the equation tan(xy) = x, we can use implicit differentiation.
First, let's differentiate both sides of the equation with respect to x:
d/dx(tan(xy)) = d/dx(x)
To differentiate the left side, we can use the chain rule. Let's break it down step by step.
1. differentiate tan(xy) with respect to xy:
d/d(xy)(tan(xy))
2. Apply the chain rule by multiplying the derivative of the outer function (tan(u)) by the derivative of the inner function (xy):
sec^2(xy) * d/d(xy)(xy)
3. Simplify by using the derivative of xy with respect to xy, which is simply 1:
sec^2(xy)
Now, let's differentiate the right side of the equation. The derivative of x with respect to x is simply 1.
So, our equation becomes:
sec^2(xy) = 1
Now, our goal is to solve for dy/dx, which represents the derivative of y with respect to x. However, at this point, we don't have enough information to solve for dy/dx. The equation we obtained, sec^2(xy) = 1, only allows us to determine the value of xy, not y itself.
To find the value of dy/dx, we need additional information or another equation involving y.
Therefore, without any further information or constraints given in the problem, we cannot determine the value of dy/dx.