function

derivative

f(x) = sinh x

f '(x) = cosh x

f(x) = cosh x

f '(x) = sinh x

f(x) = tanh x
= sinh x / cosh x 
f '(x) = sech^{ 2} x

f(x) = coth x = 1 / tanh x
= cosh x / sinh x 
f '(x) =  csch ^{ 2}x

f(x) = csch x
= 1 / sinh x

f '(x) =  csch x coth x

f(x) = sech x
= 1 / cosh x

f '(x) =  sech x tanh x

Example 1: Find the derivative of f(x) = sinh (x^{ 2})
Solution to Example 1:
 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})
Example 2: Find the derivative of f(x) = 2 sinh x + 4 cosh x
Solution to Example 2:
 Let g(x) = 2 sinh x and h(x) = 4 cosh x, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f
f '(x) = 2 cosh x + 4 sinh x
Example 3: Find the derivative of f(x) = cosh x / sinh (x^{ 2})
Solution to Example 3:
 Let g(x) = cosh x and h(x) = sinh x^{ 2}, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}, to find the derivative of function f.
g '(x) = sinh x
h '(x) = 2 x cosh x^{ 2} (see example 2 above)
f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}
= [ (sinh x^{ 2}) (sinh x)  (cosh x)(2 x cosh x^{ 2}) ] / (sinh x^{ 2})^{ 2}
Example 4: Find the derivative of f(x) = (sinh x)^{ 2}
Solution to Example 4:
 Let u = sinh x and y = u^{ 2}, Use the chain rule to find the derivative of function f as follows.
f '(x) = (dy / du) (du / dx)
 dy / du = 2 u and du / dx = cosh x
f '(x) = 2 u cosh x
 Put u = sinh x in f '(x) obtained above
f '(x) = 2 sinh x cosh x
Exercises Find the derivative of each function.
1  f(x) = sinh (x^{ 3})
2  g(x) =  sinh x + 4 cosh (x + 2)
3  h(x) = cosh x^{ 2}/ sinh x
4  j(x) =  (cosh x)^{ 2}
solutions to the above exercises
1  f '(x) = (3x^{ 2}) cosh (x^{ 3})
2  g '(x) =  cosh x + 4 sinh (x + 2)
3  h '(x) = [ (2 x sinh x^{ 2})(sinh x)  (cosh x^{ 2})(cosh x) ] / [sinh x]^{ 2}
4  j '(x) =  2 (cosh x)(sinh x)
More on differentiation and derivatives
