Differentiate tan^(-1)(cosh x) with respect to x.
Understand the Problem
The question is asking to find the derivative of the inverse tangent of the hyperbolic cosine of x with respect to x. This involves applying differentiation rules for inverse functions and hyperbolic functions.
Answer
The derivative is given by $$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
Answer for screen readers
The derivative of the inverse tangent of the hyperbolic cosine of $x$ is
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
Steps to Solve
- Identify the function to differentiate
The function we want to differentiate is $y = \tan^{-1}(\cosh(x))$.
- Use the chain rule for differentiation
To find the derivative of $y$ with respect to $x$, we apply the chain rule. The derivative of $y$ can be expressed as:
$$ \frac{dy}{dx} = \frac{d}{dx} \left( \tan^{-1}(u) \right) \cdot \frac{du}{dx} $$
where $u = \cosh(x)$.
- Find the derivative of $\tan^{-1}(u)$
The derivative of the inverse tangent function is given by:
$$ \frac{d}{du} (\tan^{-1}(u)) = \frac{1}{1 + u^2} $$
- Find the derivative of $u = \cosh(x)$
Next, we differentiate $u = \cosh(x)$. The derivative of $\cosh(x)$ is:
$$ \frac{du}{dx} = \sinh(x) $$
- Combine the derivatives
Substituting the derivatives back into the chain rule expression:
$$ \frac{dy}{dx} = \frac{1}{1 + \cosh^2(x)} \cdot \sinh(x) $$
- Simplify the derivative expression
Using the identity $\cosh^2(x) - \sinh^2(x) = 1$, we can say:
$$ 1 + \cosh^2(x) = \sinh^2(x) + 2\cosh^2(x) $$
Thus, the simplified form of the derivative becomes:
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
The derivative of the inverse tangent of the hyperbolic cosine of $x$ is
$$ \frac{dy}{dx} = \frac{\sinh(x)}{1 + \cosh^2(x)} $$
More Information
This derivative highlights the relationship between hyperbolic functions and inverse trigonometric functions. The derivation uses both the chain rule and knowledge of hyperbolic identities.
Tips
- Forget to apply chain rule: It's crucial to remember that the derivative of a composite function requires the chain rule.
- Neglect hyperbolic identities: Not using identities involving hyperbolic functions can make simplifications difficult.
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