How to Find the Derivative of the Hyperbolic Sine Function Using the Chain Rule

d/dx (sinh x)

To find the derivative of the hyperbolic sine function, denoted as sinh(x), we use the chain rule of differentiation

To find the derivative of the hyperbolic sine function, denoted as sinh(x), we use the chain rule of differentiation. The chain rule states that if we have a composition of functions, such as sinh(f(x)), then the derivative is given by the derivative of the outer function (in this case, sinh) multiplied by the derivative of the inner function (in this case, f(x)).

First, let’s start by expressing the hyperbolic sine function in terms of exponential functions:

sinh(x) = 1/2 * (e^x – e^(-x))

Now, we can differentiate this expression using the chain rule.

d/dx (sinh(x)) = d/dx [1/2 * (e^x – e^(-x))]

To differentiate this, we apply the chain rule to each term separately:

d/dx (1/2 * e^x) – d/dx (1/2 * e^(-x))

The derivative of e^x is simply e^x, and the derivative of e^(-x) is -e^(-x).

Therefore, the derivative of sinh(x) is:

d/dx (sinh(x)) = 1/2 * e^x – (-1/2 * e^(-x))

Simplifying, we obtain:

d/dx (sinh(x)) = 1/2 * (e^x + e^(-x))

So, the derivative of sinh(x) is equal to 1/2 times the sum of e^x and e^(-x).

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