Find the Derivative - d/dx sin( square root of cos(tan(pix)))

Problem

d(sin(√(,cos(tan(π*x)))))/d(x)

Solution

1. Identify the outermost function as sin(u) where u=√(,cos(tan(π*x)))

2. Apply the chain rule to differentiate sin(u) which results in cos(u)⋅d(u)/d(x)

3. Differentiate the inner square root function using the power rule, where d(√(,v))/d(x)=1/(2√(,v))⋅d(v)/d(x) and v=cos(tan(π*x))

4. Apply the chain rule to the cosine function, giving −sin(tan(π*x))⋅d(tan(π*x))/d(x)

5. Differentiate the tangent function, which results in sec2(π*x)⋅(d(π)*x)/d(x)

6. Differentiate the linear term π*x to get the constant π

7. Combine all the resulting factors from the chain rule steps into a single expression.

Final Answer

d(sin(√(,cos(tan(π*x)))))/d(x)=−(π*cos(√(,cos(tan(π*x))))*sin(tan(π*x))*sec2(π*x))/(2√(,cos(tan(π*x))))

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