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Given , find .Answer:
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Q. Given , find .Answer:
- Identify Function: Identify the function to differentiate. The function given is , which means .
- Apply Chain Rule: Recognize that to differentiate with respect to , we need to apply the chain rule because we have a function raised to a power. The chain rule states that the derivative of a composite function is .
- Differentiate Outer Function: Differentiate the outer function first, keeping the inner function the same. The outer function is where , and its derivative with respect to is . So, the derivative of the outer function with respect to is .
- Differentiate Inner Function: Now, differentiate the inner function, which is . The derivative of with respect to is , where is the cosecant function.
- Apply Chain Rule Again: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us the derivative of with respect to : .
- Multiply by Constant Factor: Finally, multiply the result by the constant factor from the original function, which is . This gives us .
- Simplify and Final Answer: Simplify the expression by combining constants and writing the final answer. The constants and can be multiplied together to give , and the negative sign remains. So, the final answer is .
