Find the 2nd Derivative sec(x-pi/2)

Problem

d2(sec(x−π/2))/(d(x)2)

Solution

1. Simplify the expression using trigonometric identities. Note that sec(x−π/2)=1/cos(x−π/2) Since cos(x−π/2)=sin(x) the expression simplifies to csc(x)

sec(x−π/2)=csc(x)

2. Find the first derivative of csc(x) with respect to x The derivative of csc(x) is −csc(x)*cot(x)

d(csc(x))/d(x)=−csc(x)*cot(x)

3. Apply the product rule to find the second derivative. The product rule states d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)′*g(x)+ƒ(x)*g(x)′ Let ƒ(x)=−csc(x) and g(x)=cot(x)

d2(csc(x))/(d(x)2)=(d(−)*csc(x))/d(x)*cot(x)+(−csc(x))d(cot(x))/d(x)

4. Differentiate the individual components. The derivative of −csc(x) is csc(x)*cot(x) and the derivative of cot(x) is −csc2(x)

d2(csc(x))/(d(x)2)=(csc(x)*cot(x))*cot(x)+(−csc(x))*(−csc2(x))

5. Simplify the resulting expression by combining terms.

d2(csc(x))/(d(x)2)=csc(x)*cot2(x)+csc3(x)

6. Factor out the common term csc(x) to reach the final form.

d2(csc(x))/(d(x)2)=csc(x)*(cot2(x)+csc2(x))

Final Answer

d2(sec(x−π/2))/(d(x)2)=csc(x)*(cot2(x)+csc2(x))

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