Differentiate √(1+sinh²x) with respect to x
Understand the Problem
The question is asking to find the derivative of the function √(1+sinh²x) with respect to the variable x. To solve this, we will use differentiation rules, including the chain rule and the derivative of hyperbolic sine.
Answer
$$ \frac{\sinh{x}\cosh{x}}{\sqrt{1+\sinh^2{x}}} $$
Answer for screen readers
The derivative of the function ( \sqrt{1+\sinh^2{x}} ) with respect to ( x ) is
$$ \frac{\sinh{x}\cosh{x}}{\sqrt{1+\sinh^2{x}}} $$
Steps to Solve
- Apply the derivative of a square root
To find the derivative of the function ( \sqrt{1+\sinh^2{x}} ), we start by using the chain rule. The derivative of ( \sqrt{u} ) is ( \frac{1}{2\sqrt{u}} ), where ( u = 1 + \sinh^2{x} ).
So, we have:
$$ \frac{d}{dx} \sqrt{1+\sinh^2{x}} = \frac{1}{2\sqrt{1+\sinh^2{x}}} \cdot \frac{d}{dx}(1+\sinh^2{x}) $$
- Differentiate the inner function
Now we need to differentiate ( 1+\sinh^2{x} ). The constant 1 contributes 0, so we differentiate ( \sinh^2{x} ) using the chain rule again: [ \frac{d}{dx}(\sinh^2{x}) = 2\sinh{x} \cdot \cosh{x} ]
Thus, the derivative is
$$ \frac{d}{dx}(1+\sinh^2{x}) = 0 + 2\sinh{x} \cdot \cosh{x} = 2\sinh{x}\cosh{x} $$
- Combine the results
Now, we can combine our derivatives:
$$ \frac{d}{dx} \sqrt{1+\sinh^2{x}} = \frac{1}{2\sqrt{1+\sinh^2{x}}} \cdot 2\sinh{x}\cosh{x} $$
This simplifies to:
$$ \frac{\sinh{x}\cosh{x}}{\sqrt{1+\sinh^2{x}}} $$
The derivative of the function ( \sqrt{1+\sinh^2{x}} ) with respect to ( x ) is
$$ \frac{\sinh{x}\cosh{x}}{\sqrt{1+\sinh^2{x}}} $$
More Information
The derivative found represents how the function ( \sqrt{1+\sinh^2{x}} ) changes as ( x ) changes. The use of hyperbolic functions is common in calculus, particularly when modeling growth processes or in problems involving areas and distances in different geometries.
Tips
- Forgetting to apply the chain rule properly when differentiating composite functions.
- Not simplifying the expression after applying the derivative, which can lead to unnecessary complexity.
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