Use implicit differentiation to find dy/dx.
x^5 = cot y
To find the derivative dy/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x and then solve for dy/dx.
Starting with the given equation:
x^5 = cot(y)
Differentiating both sides with respect to x:
(d/dx) (x^5) = (d/dx) (cot(y))
Now let's evaluate each derivative separately:
The derivative of x^5 with respect to x can be found using the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1). Applying this formula, we have:
(d/dx) (x^5) = 5x^4
To differentiate cot(y) with respect to x, we need to use the chain rule. The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
In this case, y = cot(y) and u = y. So, applying the chain rule, we have:
(d/dx) (cot(y)) = (dy/du) * (du/dx)
The derivative of cot(y) with respect to u (dy/du) can be found using the derivative of cot(u) with respect to u, which is -csc^2(u). Since our u is y, we have:
(dy/du) = -csc^2(y)
The derivative du/dx is simply dy/dx since u = y.
Putting it all together, we have:
5x^4 = -csc^2(y) * dy/dx
To solve for dy/dx, divide both sides by -csc^2(y):
dy/dx = 5x^4 / -csc^2(y)
Therefore, the derivative dy/dx is equal to 5x^4 / -csc^2(y).