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Second derivative of sinh⁡(x)\sinh{\left(x \right)}

The calculator will find the second derivative of sinh⁡(x)\sinh{\left(x \right)}, with steps shown.

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Your Input

Find d2dx2(sinh⁡(x))\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right).

Solution

Find the first derivative ddx(sinh⁡(x))\frac{d}{dx} \left(\sinh{\left(x \right)}\right)

The derivative of the hyperbolic sine is ddx(sinh⁡(x))=cosh⁡(x)\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}:

(ddx(sinh⁡(x)))=(cosh⁡(x)){\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}

Thus, ddx(sinh⁡(x))=cosh⁡(x)\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}.

Next, d2dx2(sinh⁡(x))=ddx(cosh⁡(x))\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \frac{d}{dx} \left(\cosh{\left(x \right)}\right)

The derivative of the hyperbolic cosine is ddx(cosh⁡(x))=sinh⁡(x)\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}:

(ddx(cosh⁡(x)))=(sinh⁡(x)){\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}

Thus, ddx(cosh⁡(x))=sinh⁡(x)\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}.

Therefore, d2dx2(sinh⁡(x))=sinh⁡(x)\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \sinh{\left(x \right)}.

Answer

d2dx2(sinh⁡(x))=sinh⁡(x)\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \sinh{\left(x \right)}A