Differentiate arctan(cosh(x)) with respect to x.
Understand the Problem
The question is asking to differentiate the function arctan(cosh(x)) with respect to x. This involves applying the chain rule and the derivative of the arctangent function as part of the differentiation process.
Answer
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
Answer for screen readers
The derivative of the function ( y = \arctan(\cosh(x)) ) with respect to ( x ) is:
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
Steps to Solve
- Identify the function to differentiate
We are given the function ( y = \arctan(\cosh(x)) ).
- Apply the chain rule
To find the derivative ( \frac{dy}{dx} ), we use the chain rule. The derivative of ( \arctan(u) ) is ( \frac{1}{1 + u^2} ), where ( u = \cosh(x) ). So we have:
$$ \frac{dy}{dx} = \frac{1}{1 + \cosh^2(x)} \cdot \frac{d}{dx}(\cosh(x)) $$
- Differentiate cosh(x)
The derivative of ( \cosh(x) ) is ( \sinh(x) ). Therefore:
$$ \frac{d}{dx}(\cosh(x)) = \sinh(x) $$
- Combine the results
Now, substitute ( \frac{d}{dx}(\cosh(x)) ) back into the equation for ( \frac{dy}{dx} ):
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
- Simplify if necessary
No further simplification is needed since it presents the derivative clearly.
The derivative of the function ( y = \arctan(\cosh(x)) ) with respect to ( x ) is:
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
More Information
The function ( \arctan ) is commonly used in calculus, and differentiating it involves recognizing how the inner function (in this case ( \cosh(x) )) interacts with it. Knowing the derivatives of hyperbolic functions like ( \sinh ) and ( \cosh ) is useful in these situations.
Tips
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Confusing the derivatives of trigonometric and hyperbolic functions. Ensure to remember ( \frac{d}{dx}(\sinh(x)) = \cosh(x) ) and ( \frac{d}{dx}(\cosh(x)) = \sinh(x) ).
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Forgetting to apply the chain rule correctly. Each layer of function needs to be differentiated.
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