Find dy/dx x=tan(y)

Problem

x=tan(y)

Solution

1. Differentiate both sides of the equation with respect to x using implicit differentiation.

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

2. Apply the chain rule to the right side of the equation, noting that y is a function of x

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

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

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

4. Use the trigonometric identity sec2(y)=1+tan2(y) to express the derivative in terms of y

d(y)/d(x)=1/(1+tan2(y))

5. Substitute the original relationship x=tan(y) back into the expression to write the derivative in terms of x

d(y)/d(x)=1/(1+x2)

Final Answer

d(y)/d(x)=1/(1+x2)

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