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function
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derivative
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f(x) = sinh x
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f '(x) = cosh x
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f(x) = cosh x
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f '(x) = sinh x
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f(x) = tanh x
= sinh x / cosh x |
f '(x) = sech 2 x
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f(x) = coth x = 1 / tanh x
= cosh x / sinh x |
f '(x) = - csch 2x
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f(x) = csch x
= 1 / sinh x
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f '(x) = - csch x coth x
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f(x) = sech x
= 1 / cosh x
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f '(x) = - sech x tanh x
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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
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