Find the 2nd Derivative y=tan(x)

Problem

d2(tan(x))/(d(x)2)

Solution

1. Identify the function to be differentiated, which is y=tan(x)

2. Find the first derivative by applying the standard trigonometric derivative rule d(tan(x))/d(x)=sec2(x)

d(y)/d(x)=sec2(x)

3. Apply the chain rule to find the second derivative, treating sec2(x) as (sec(x))2

d2(y)/(d(x)2)=d(sec(x))/d(x)

4. Differentiate the outer power function and multiply by the derivative of the inner function sec(x)

d2(y)/(d(x)2)=2*sec(x)⋅d(sec(x))/d(x)

5. Substitute the derivative d(sec(x))/d(x)=sec(x)*tan(x) into the expression.

d2(y)/(d(x)2)=2*sec(x)⋅sec(x)*tan(x)

6. Simplify the expression by combining the sec(x) terms.

d2(y)/(d(x)2)=2*sec2(x)*tan(x)

Final Answer

d2(tan(x))/(d(x)2)=2*sec2(x)*tan(x)

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