What is the d/(dt)(at-tanh(at)) ?

The d/(dt)(at-tanh(at)) is atanh^2(at)

What is the first d/(dt)(at-tanh(at)) ?

The first d/(dt)(at-tanh(at)) is atanh^2(at)

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\frac{d}{d t} (a t - tanh (a t))

a tanh^{2} (a t)

Solution steps

\frac{d}{d t} (a t - tanh (a t))

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d t} (a t) - \frac{d}{d t} (tanh (a t))

\frac{d}{d t} (a t) = a

\frac{d}{d t} (tanh (a t)) = sech^{2} (a t) a

= a - sech^{2} (a t) a

Simplify

a - sech^{2} (a t) a : a tanh^{2} (a t)

= a tanh^{2} (a t)

What is the d/(dt)(at-tanh(at)) ?
The d/(dt)(at-tanh(at)) is atanh^2(at)
What is the first d/(dt)(at-tanh(at)) ?
The first d/(dt)(at-tanh(at)) is atanh^2(at)