Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

What is the derivative of tanh(x)?

Answer
VerifiedVerified
451.8k+ views
like imagedislike image
Hint: In this question we have to find the derivative of the hyperbolic function. In mathematics, the hyperbolic functions are similar to the trigonometric functions or circular functions. The hyperbolic functions are analogs of the circular function or the trigonometric functions. The basic hyperbolic functions are hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh).

Complete step by step solution:
Hyperbolic functions work in the same way as the “normal” trigonometric “cuisine” but instead of referring to a unit circle, they refer to a set of hyperbolas. If we compare the derivatives of hyperbolic functions with the derivatives of the standard trigonometric functions, then we find lots of similarities and differences also. For example, the derivatives of the sine functions match:
ddxsinx=cosxddxsinhx=coshx
But we can see difference also
ddxcosx=sinxddxcoshx=sinhx
The hyperbolic functions are defined through the algebraic expression that includes the exponential function ex and its inverse exponential functionex. To find the derivative of the tanhx, we will use the trigonometric rule which is sinhxcoshx=tanhx.
Now the value of sinhx in term of exponential function is:
sinhx=12(exex)
Similarly, the value of coshx in term of exponential function is:
coshx=12(ex+ex)
Now,
tanhx=sinhxcoshxtanhx=12(exex)12(ex+ex)tanhx=(exex)(ex+ex)
Now we will differentiate it with respect to x. since we can easily see that the above expression will be differentiated by quotient rule.
We know the quotient rule is:
d(uv)dx=dudxvdvdxuv2
We also know the derivative of ex is ex, and derivative of ex is ex .
Now applying the quotient rule, we get
ddx(tanhx)=(exex) `(ex+ex)(exex)(ex+ex) `(ex+ex)2ddx(tanhx)=(ex+ex)(ex+ex)(exex)(exex)(ex+ex)2
Now by more simplifying, we get
ddx(tanhx)=((ex+ex)2(exex)2)(ex+ex)2ddx(tanhx)=1(exex)2(ex+ex)2
Now we can write tanhx=exexex+ex
ddx(tanhx)=1tanh2x
Since the hyperbolic function identities are similar to the trigonometric functions, like cosh2x+sinh2x=1,1tanh2x=sech2x,coth2xcosech2x=1 .
So, we can also write the derivative of the tanhx as ddx(tanhx)=1tanh2x=sech2x.
Hence we get the derivative of the tanhx which is sech2x

Note: The hyperbolic functions are used to define distance in specific non- Euclidean geometry, which means estimating the angles and distances in hyperbolic geometry. The basic difference between trigonometric functions and hyperbolic functions is that trigonometric functions can be defined with the rotations along a circle, while hyperbolic functions can be defined with the use of rotations along a hyperbola.

0 views