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.