Find the Derivative Using Chain Rule - d/dz y=cot(sin(x))^2

Problem

d()/d(x)*cot2(sin(x))

Solution

1. Identify the outer function and the inner functions to apply the chain rule. The expression is of the form u2 where u=cot(sin(x))

2. Apply the power rule to the outermost part, which is the square.

d()/d(x)*u2=2*u⋅d(u)/d(x)

3. Differentiate the cotangent function using the chain rule again. The derivative of cot(v) is −csc2(v)⋅d(v)/d(x) where v=sin(x)

d(cot(sin(x)))/d(x)=−csc2(sin(x))⋅d(sin(x))/d(x)

4. Differentiate the innermost function, which is sin(x)

d(sin(x))/d(x)=cos(x)

5. Combine all the parts using the chain rule.

d(y)/d(x)=2*cot(sin(x))⋅(−csc2(sin(x)))⋅cos(x)

6. Simplify the expression by rearranging the terms and signs.

d(y)/d(x)=−2*cos(x)*cot(sin(x))*csc2(sin(x))

Final Answer

d()/d(x)*cot2(sin(x))=−2*cos(x)*cot(sin(x))*csc2(sin(x))

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