Example 1: Evaluate
Solution: Use the quotient rule
where u = x^{6} and v = sinh(x) +1
The derivative of a sum is the sum of the derivatives.
The derivative of the constant 1 is 0.
The derivative of sinh(x) is cosh(x)
The derivative of x^{6} is 6x^{6-1
}
Simplify the equation we get:
Example 2: Differentiate each of the following functions
a) f(x) = 2x^{5}cosh(x)
b) h(t) = sinh(t)/(t+1)
Solution:
a) f´(x) = 10x^{4}cosh(x) + 2x^{5}sinh(x)
b)
Example 3: Find the derivative of f(x) = sinh (x^{ 2})
Solution :
Let u = x^{ 2} and y = sinh u and use the chain rule to find the derivative of the given function f as follows.
f '(x) = (dy / du) (du / dx)
dy / du = cosh u, see formula above, and du / dx = 2 x
f '(x) = 2 x cosh u = 2 x cosh (x^{ 2})
Substitute u = x^{ 2} in f '(x) to obtain
f '(x) = 2 x cosh (x^{ 2})