1 answer:
Given:
![sinh(f(x))=1+x^2](https://tex.z-dn.net/?f=sinh%28f%28x%29%29%3D1%2Bx%5E2)
To find:
The value of f'(x).
Solution:
Formulae used:
![\dfrac{d}{dx}sinh(x)=cosh(x)](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7Dsinh%28x%29%3Dcosh%28x%29)
![\dfrac{d}{dx}x^n=nx^{n-1}](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7Dx%5En%3Dnx%5E%7Bn-1%7D)
![\dfrac{d}{dx}C=0](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7DC%3D0)
Chain rule:
![\dfrac{d}{dx}[f(g(x))]=f'(g(x))g'(x)](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%3Df%27%28g%28x%29%29g%27%28x%29)
Where C is an arbitrary constant.
We have,
![sinh(f(x))=1+x^2](https://tex.z-dn.net/?f=sinh%28f%28x%29%29%3D1%2Bx%5E2)
Differentiate with respect to x.
![cosh(f(x))f'(x)=0+2x](https://tex.z-dn.net/?f=cosh%28f%28x%29%29f%27%28x%29%3D0%2B2x)
![f'(x)=\dfrac{2x}{cosh(f(x))}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cdfrac%7B2x%7D%7Bcosh%28f%28x%29%29%7D)
Therefore, the required values is
.
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