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Find the derivative of .
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Q. Find the derivative of .
- Identify Outer Function: Identify the outer function and its derivative.The outer function is the cosine function, so we need to find the derivative of , where is a function of . The derivative of with respect to is .
- Identify Inner Function: Identify the inner function and its derivative.The inner function is . To find its derivative, we use the chain rule. The derivative of with respect to is , so the derivative of is . Here, , so the derivative of the inner function is .
- Apply Chain Rule: Apply the chain rule.The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. So, we have:
- Differentiate Inner Function: Differentiate the inner function.We already found that the derivative of is . Now we need to differentiate with respect to , which is simply . So the derivative of the inner function is .
- Simplify Inner Function Derivative: Simplify the derivative of the inner function.Multiplying by gives us , which simplifies to because the negatives cancel out.
- Combine Derivatives: Combine the derivatives to find the final derivative.Now we multiply the derivative of the outer function by the simplified derivative of the inner function:
- Simplify Final Expression: Simplify the final expression.We can leave the final derivative in its current form, as it is already simplified:
