Derivative of sinhx: Formula, Proof | sinhx Derivative

The derivative of sinhx, denoted by d/dx(sinhx), is equal to coshx. In this post, we will learn how to differentiate sinh(x), i.e, how to find the derivative of the hyperbolic sine function with respect to x.

The derivative formula of sinhx is given as follows:

d/dx[sinhx] = coshx.

Before differentiating sinh(x), let us first recall that

1. sinh x = (e^x – e^-x)/2
2. cosh x = (e^x + e^-x)/2

Find the Derivative of sinhx

We know that

$\sinh x=\dfrac{e^x-e^{-x}}{2}$.

Differentiating both sides with respect to x, the derivative of sinh x will be equal to

$\dfrac{d}{dx}\left(\sinh x \right )$ $=\dfrac{d}{dx}\left(\dfrac{e^x -e^{-x}}{2}\right)$

⇒$\dfrac{d}{dx}\left(\sinh x \right )$ = $\dfrac{1}{2} \Big[\dfrac{d}{dx}\left(e^x \right) – \dfrac{d}{dx}\left(e^{-x}\right) \Big]$ by the linearity of derivatives.

⇒$\dfrac{d}{dx}\left(\sinh x \right )$ = $\dfrac{1}{2} \cdot$ [e^x – (-e^-x)] as we know that $\dfrac{d}{dx}\left(e^{mx} \right)=me^{mx}$

= (e^x + e^-x)/2

= cosh(x), by the above Formula (2).

Hence the derivative of sinh(x) is equal to cosh(x).

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