Deriving The Formula For The Derivative Of Cosh^-1 X Using Chain Rule And Hyperbolic Functions

d/dx (cosh^-1 x)

1 / √(x^2 – 1)

We can start the problem by using the formula d/dx (cosh^-1 x) = 1 / sqrt(x^2 – 1). This formula can be derived using the chain rule and the definition of hyperbolic functions.

To prove this formula, we can start by using the definition of hyperbolic cosine inverse:

cosh^-1 x = ln(x + sqrt(x^2 – 1))

Then, using the chain rule, we can find the derivative:

d/dx (cosh^-1 x) = d/dx (ln(x + sqrt(x^2 – 1)))

Using the chain rule again, we can simplify this expression:

d/dx (cosh^-1 x) = 1 / (x + sqrt(x^2 – 1)) * (1 + (x / sqrt(x^2 – 1)))

Now, we can simplify this expression further by multiplying the numerator and denominator of the second term by (sqrt(x^2 – 1) – x), which will cancel out the square root:

d/dx (cosh^-1 x) = 1 / (x + sqrt(x^2 – 1)) * (1 + (x / sqrt(x^2 – 1))) * ((sqrt(x^2 – 1) – x) / (sqrt(x^2 – 1) – x))

d/dx (cosh^-1 x) = (sqrt(x^2 – 1) – x) / (x^2 – 1)

Finally, we can simplify this expression using the identity x^2 – 1 = (x + 1)(x – 1):

d/dx (cosh^-1 x) = 1 / sqrt(x^2 – 1)

Therefore, the derivative of cosh^-1 x is 1 / sqrt(x^2 – 1).

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