Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
10. $\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}$
To state:
Whether the given statement,"d2ydx2=dydx2" is true of false.
False.
Counter example,
fx=x2
From definition
f'x=limh→0fx+h-fxh
=limh→0x+h2-x2h
=limh→0(x2+2xh+h2)-x2h
=limh→02xh+h2h
=limh→0h(2x+h)h
=limh→0 2x+h=2x
f'x=2x
[f'x]2=4x2 …(1)
f''x=limh→0f'x+h-f'xh
=limh→02x+h-2xh
=limh→02hh=limh→02=2
f''x=2 …(2)
From (1) and (2)
f''x≠[f'x]2
Final statement:
d2ydx2=dydx2 is false statement.