To determine

To find: The second order derivative of f(t).

Answer

The second order derivative of f(t) is f″(π4)=32.

Explanation

Given:

The function is f(t)=sect.

Formula used:

Product Rule:

If f(t) and g(t) are differentiable function, then the product rule is,

ddt[f(t)g(t)]=f(t)ddt[g(t)]+g(t)ddt[f(t)] (1)

Power Rule:

If n is positive integer, then the power rule is,

ddx(xn)=nxn−1 (2)

Calculation:

Obtain second order derivative of f(t).

ddt[sect]=secttant

Apply Product Rule as shown in equation (1).

ddt[secttant]=sectddt[tant]+tantddt[sect]f″(t)=sect(sec2t)+tant(secttant)f″(t)=sec3t+secttan2t

Substitute π4 for t,

f″(π4)=(secπ4)3+secπ4(tanπ4)2=(2)3+2(1)2      {∵secπ4=2,tanπ4=1}=222+2=2(22+1)

Simplify the equation.

 f″(π4)=2(2+1)=32

Thus, the second order derivative of f(t)=sect is f″(π4)=32.