cot(y) = x − 4y
Find using implicit differentiation
coty = x - 4y
-csc^2y y' = 1 - 4y'
(4-csc^2y)y' = 1
y' = 1/(4-csc^2y)
To find the derivative of cot(y) = x - 4y with respect to x using implicit differentiation, you need to follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Step 2: Apply the chain rule for the derivative of cot(y), which is -csc^2(y)*dy/dx.
Step 3: Solve the resulting equation for dy/dx to find the derivative.
Let's go through each step in detail.
Step 1: Differentiate both sides of the equation
Differentiating cot(y) = x - 4y with respect to x gives:
-d/dx[cot(y)] = d/dx[x - 4y]
Step 2: Apply the chain rule for cot(y)
The derivative of cot(y) with respect to x can be written as -csc^2(y) * dy/dx. So the left side of the equation becomes:
-csc^2(y) * dy/dx
Step 3: Solve for dy/dx
Now we can substitute back into the equation and solve for dy/dx:
-csc^2(y) * dy/dx = 1 - 4(dy/dx)
Next, we can group the terms with dy/dx together:
(dy/dx) * (-csc^2(y) + 4) = 1
Finally, we isolate dy/dx by dividing both sides by (-csc^2(y) + 4):
dy/dx = 1 / (-csc^2(y) + 4)
Therefore, the derivative of cot(y) = x - 4y with respect to x using implicit differentiation is dy/dx = 1 / (-csc^2(y) + 4).