Simplifying the Derivative of Csch x With Respect to x Using the Quotient Rule

d/dx (csch x)

To find the derivative of csch x with respect to x, we can use the quotient rule

To find the derivative of csch x with respect to x, we can use the quotient rule. The formula for the derivative of csch x is:

d/dx (csch x) = -csch x * coth x

Here’s a step-by-step explanation of how we get to this result:

Step 1: Start by writing the function in terms of sinh x:

csch x = 1/sinh x

Step 2: Apply the quotient rule to differentiate the function:

d/dx (csch x) = (d/dx (1))/(sinh x) – (1/(sinh x))(d/dx (sinh x))

Step 3: Differentiate the numerator:

d/dx (1) = 0

Step 4: Differentiate sinh x using the chain rule:

d/dx (sinh x) = cosh x

Step 5: Substitute these results back into the quotient rule formula:

d/dx (csch x) = 0 – (1/(sinh x))(cosh x)

Step 6: Simplify the expression:

d/dx (csch x) = -cosh x / sinh x

Since cosh x / sinh x is equivalent to coth x, we can rewrite the expression as:

d/dx (csch x) = -csch x * coth x

This is the final answer for the derivative of csch x with respect to x.

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