Find the Derivative - d/dx tan(xy)

Problem

d(tan(x*y))/d(x)

Solution

1. Apply the chain rule to the outer function, which is the tangent function.

d(tan(u))/d(u)=sec2(u)

2. Differentiate the inner function x*y with respect to x using the product rule.

(d(x)*y)/d(x)=xd(y)/d(x)+yd(x)/d(x)

3. Simplify the derivative of the inner function.

(d(x)*y)/d(x)=xd(y)/d(x)+y

4. Combine the results by multiplying the derivative of the outer function by the derivative of the inner function.

d(tan(x*y))/d(x)=sec2(x*y)*(xd(y)/d(x)+y)

Final Answer

d(tan(x*y))/d(x)=sec2(x*y)*(xd(y)/d(x)+y)

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