Find the Derivative - d/dx y=tan(cos(x))

Problem

d()/d(x)*tan(cos(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=cos(x)

2. Apply the Chain Rule by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

d(tan(u))/d(u)⋅d(u)/d(x)

3. Differentiate the outer function tan(u) which results in sec2(u)

sec2(cos(x))⋅d(cos(x))/d(x)

4. Differentiate the inner function cos(x) which results in −sin(x)

sec2(cos(x))⋅(−sin(x))

5. Simplify the expression by moving the negative sign and the sine function to the front.

−sin(x)*sec2(cos(x))

Final Answer

d(tan(cos(x)))/d(x)=−sin(x)*sec2(cos(x))

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