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Second derivative of $$$\sinh{\left(x \right)}$$$
Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator
Your Input
Find $$$\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right)$$$.
Solution
Find the first derivative $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right)$$$
The derivative of the hyperbolic sine is $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}$$Thus, $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$.
Next, $$$\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \frac{d}{dx} \left(\cosh{\left(x \right)}\right)$$$
The derivative of the hyperbolic cosine is $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}$$Thus, $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$.
Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$.
Answer
$$$\frac{d^{2}}{dx^{2}} \left(\sinh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$A