Find the Derivative - d/dx tan( square root of 1-x)

Problem

d(tan(√(,1−x)))/d(x)

Solution

1. Identify the outer function and the inner function to apply the chain rule. The outer function is tan(u) and the inner function is u=√(,1−x)

2. Differentiate the outer function with respect to its argument. The derivative of tan(u) is sec2(u)

sec2(√(,1−x))

3. Apply the chain rule by multiplying by the derivative of the inner function √(,1−x) To differentiate √(,1−x) use the power rule on (1−x)(1/2)

d(√(,1−x))/d(x)=1/2*(1−x)(−1/2)⋅d(1−x)/d(x)

4. Differentiate the innermost expression (1−x) The derivative is −1

d(√(,1−x))/d(x)=1/(2√(,1−x))⋅(−1)

5. Combine all parts of the chain rule together.

d(tan(√(,1−x)))/d(x)=sec2(√(,1−x))⋅(−1)/(2√(,1−x))

6. Simplify the final expression by moving the constant and the denominator to the front.

−sec2(√(,1−x))/(2√(,1−x))

Final Answer

d(tan(√(,1−x)))/d(x)=−sec2(√(,1−x))/(2√(,1−x))

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