## Proof

The hyperbolic sine function is defined as:

$$\mathrm{sinh}(x)=\frac{{e}^{x}-{e}^{-x}}{2}$$To find the derivative, we use the derivative of exponential functions. The derivative of ${e}^{x}$ is ${e}^{x}$, and the derivative of ${e}^{-x}$ is $-{e}^{-x}$.

Now, differentiating $\mathrm{sinh}(x)$:

$$\frac{\text{d}}{\text{d}x}\left(\frac{{e}^{x}-{e}^{-x}}{2}\right)=\frac{1}{2}({e}^{x}-(-{e}^{-x}))$$This simplifies to:

$$\frac{1}{2}({e}^{x}+{e}^{-x})=\mathrm{cosh}(x)$$Thus, the derivative of $\mathrm{sinh}(x)$ is:

$$\frac{\text{d}}{\text{d}x}\mathrm{sinh}(x)=\mathrm{cosh}(x)$$## Explanation

The hyperbolic sine function, $\mathrm{sinh}(x)$, is similar to the sine function but based on exponential functions. It is defined as:

$$\mathrm{sinh}(x)=\frac{{e}^{x}-{e}^{-x}}{2}$$This expression represents the difference between the exponential growth ${e}^{x}$ and the exponential decay ${e}^{-x}$, divided by two.

To find the derivative, we differentiate each part of the function. The derivative of ${e}^{x}$ with respect to $x$ is ${e}^{x}$, and the derivative of ${e}^{-x}$ with respect to $x$ is $-{e}^{-x}$. This is because of the chain rule, where the derivative of $-x$ is $-1$.

Substituting these derivatives into the formula for $\mathrm{sinh}(x)$:

$$\frac{\text{d}}{\text{d}x}\left(\frac{{e}^{x}-{e}^{-x}}{2}\right)=\frac{1}{2}({e}^{x}-(-{e}^{-x}))$$This simplifies to:

$$\frac{1}{2}({e}^{x}+{e}^{-x})$$This expression is exactly the definition of $\mathrm{cosh}(x)$, the hyperbolic cosine function:

$$\frac{{e}^{x}+{e}^{-x}}{2}\phantom{\rule{0.167em}{0ex}}=\phantom{\rule{0.167em}{0ex}}\overline{)\mathrm{cosh}(x)}$$*Q.E.D.*