Differentiate coth(tan(x)) with respect to x
Understand the Problem
The question is asking us to differentiate the function coth(tan(x)) with respect to x. To solve this, we will apply the chain rule in differentiation, which states that the derivative of a composite function can be found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Answer
The derivative is \( \frac{dy}{dx} = -\csch^2(\tan(x)) \cdot \sec^2(x) \).
Answer for screen readers
The derivative of ( \coth(\tan(x)) ) with respect to ( x ) is: $$ \frac{dy}{dx} = -\csch^2(\tan(x)) \cdot \sec^2(x) $$
Steps to Solve
- Identify the functions involved
Let ( y = \coth(u) ) where ( u = \tan(x) ). We need to differentiate ( y ) with respect to ( x ).
- Differentiate the outer function
The derivative of ( \coth(u) ) with respect to ( u ) is: $$ \frac{dy}{du} = -\csch^2(u) $$
- Differentiate the inner function
Now, we differentiate ( u ) with respect to ( x ): $$ u = \tan(x) $$ Thus, the derivative is: $$ \frac{du}{dx} = \sec^2(x) $$
- Apply the chain rule
Using the chain rule, we combine the derivatives: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
- Substitute the derivatives
Now, substitute the derivatives we found: $$ \frac{dy}{dx} = -\csch^2(\tan(x)) \cdot \sec^2(x) $$
The derivative of ( \coth(\tan(x)) ) with respect to ( x ) is: $$ \frac{dy}{dx} = -\csch^2(\tan(x)) \cdot \sec^2(x) $$
More Information
The function ( \coth(x) ) is the hyperbolic cotangent function, and its derivative involves the hyperbolic cosecant function ( \csch(x) ). Understanding the derivatives of hyperbolic functions is crucial in calculus.
Tips
- Forgetting to apply the chain rule when differentiating composite functions. Always ensure you differentiate both the outer and inner functions.
- Miscalculating the derivatives of hyperbolic functions. It's important to remember the specific rules for these derivatives.
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