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Math Help - tanh differentiable

  1. #1
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    tanh differentiable

    tanh(x)= \frac{e^x- e^{-x}}{e^x+ e^{-x}}
    (a) Prove that tanh is differentiable.

    exp is differentiable, therefore tanh is differentiable?? The above has substraction and division by exp. How do i explain that tanh is still differentiable.

    (b) Prove that tanh is strictly increasing, and hence invertible

    the derivative of tanh is greater than zero, therefore tanh is strictly increasing. How do i prove its invertible.

    Many thanks for any help.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by charikaar View Post
    How do I prove that tanh is differentiable?
    By differentiating it.
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  3. #3
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    Quote Originally Posted by charikaar View Post
    tanh(x)= \frac{e^x- e^{-x}}{e^x+ e^{-x}}
    (a) Prove that tanh is differentiable.

    exp is differentiable, therefore tanh is differentiable?? The above has substraction and division by exp. How do i explain that tanh is still differentiable.
    \frac{d}{dx} [tanh(x)] = \frac{d}{dx} \frac{sinh(x)}{cosh(x)} =\frac{cosh(x) \frac{d}{dx} [sinh(x)]-sinh(x)\frac{d}{dx}[cosh(x)]}{cosh^2 (x)}

    = \frac{cosh^2(x)-sinh^2(x)}{cosh^2(x)}=\frac{1}{cosh^2(x)} = sech^2(x)

    You see it's easy to prove that it is differentiable, isn't it? You just need to know the approperiate identities.
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