Find Where dy/dx is Equal to Zero x=tan(y)

Problem

x=tan(y)

Solution

1. Differentiate implicitly with respect to x on both sides of the equation.

d(x)/d(x)=d(tan(y))/d(x)

2. Apply the chain rule to the right side, noting that the derivative of tan(y) is sec2(y)d(y)/d(x)

1=sec2(y)d(y)/d(x)

3. Solve for the derivative d(y)/d(x) by dividing both sides by sec2(y)

d(y)/d(x)=1/sec2(y)

4. Simplify the expression using the trigonometric identity 1/sec2(y)=cos2(y)

d(y)/d(x)=cos2(y)

5. Set the derivative to zero to find the values of y where the slope is zero.

cos2(y)=0

6. Solve for y by taking the square root and finding where the cosine function equals zero.

cos(y)=0

y=π/2+n*π

7. Determine the x-values by substituting these y values back into the original equation x=tan(y)

x=tan(π/2+n*π)

8. Identify the behavior of the function. Since tan(y) is undefined at y=π/2+n*π there are no real values of x where the derivative is equal to zero.

Final Answer

d(y)/d(x)=0⇒No solution

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