d/dx cosh x
Understand the Problem
The question is asking for the derivative of the hyperbolic cosine function, cosh x, with respect to x. To solve this, we will apply the rules of differentiation specific to hyperbolic functions.
Answer
The derivative of \( \cosh x \) is \( \sinh x \).
Answer for screen readers
The derivative of the hyperbolic cosine function is ( \sinh x ).
Steps to Solve
- Identify the hyperbolic cosine function
The hyperbolic cosine function is represented as $y = \cosh(x)$.
- Recall the derivative formula
The derivative of the hyperbolic cosine function with respect to $x$ is given by the known derivative formula: $$ \frac{d}{dx} (\cosh x) = \sinh x $$
- Apply the derivative formula
Now, we can apply the derivative formula directly to our function: $$ \frac{dy}{dx} = \sinh(x) $$
- State the result
Thus, the derivative of the hyperbolic cosine function is confirmed as: $$ \frac{d}{dx} (\cosh x) = \sinh x $$
The derivative of the hyperbolic cosine function is ( \sinh x ).
More Information
The hyperbolic sine function, denoted as ( \sinh x ), is defined similarly to cosine and plays a crucial role in hyperbolic geometry and many areas of mathematics and physics.
Tips
- A common mistake is to confuse the derivatives of hyperbolic functions with their trigonometric counterparts. Remember that ( \frac{d}{dx}(\cos x) = -\sin x ) but for hyperbolic functions this is different: ( \frac{d}{dx}(\cosh x) = \sinh x ).
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