Example 4 Video Example
Prove that -(sinh-l(x)) =
dx
√1+x²
Solution 1
Let y
sinh (x). Then sinh(y) = x. If we differentiate this equation implicitly with respect to x, we get
1.
dx
Since cosh?() - sinh?(y) = 1 and cosh(y) 2 0, we have cosh(y) - 1 + sinh?(y), so
1
dy
dx
1
cosh(y)
V1 + sinh?v)
Solution 2
we have
From the equation sinh-+(x) = \n(x + V x2 + 1).
-(sinh 'x) - 4 n(x + 2 + 1)-- --4(x + x + 1)
dx
- ** V*2+1 ( + vota)-
√²+1+x
1+1
+ 1
1