Find the 2nd Derivative tan(x)

Problem

tan(x)

Solution

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

2. Find the first derivative by applying the standard trigonometric differentiation rule for tan(x)

d(tan(x))/d(x)=sec2(x)

3. Rewrite the first derivative to prepare for the second differentiation using the power rule and chain rule.

sec2(x)=(sec(x))2

4. Apply the chain rule to find the second derivative, treating sec(x) as the inner function.

d(sec(x))/d(x)=2*sec(x)⋅d(sec(x))/d(x)

5. Substitute the derivative of sec(x) which is sec(x)*tan(x)

d2(tan(x))/(d(x)2)=2*sec(x)⋅sec(x)*tan(x)

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

d2(tan(x))/(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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