Differentiate tan^(-1)(sech(x^2)) with respect to x^2.
Understand the Problem
The question is asking for the differentiation of the function tan^{-1}(sech(x^2)) with respect to x^2. This involves applying the chain rule and recognizing the appropriate derivative of the inverse tangent and the hyperbolic secant function.
Answer
$$ \frac{d}{dx^2} \tan^{-1}(\text{sech}(x^2)) = \frac{-\text{sech}(x^2) \tanh(x^2)}{1 + (\text{sech}(x^2))^2} $$
Answer for screen readers
$$ \frac{d}{dx^2} \tan^{-1}(\text{sech}(x^2)) = \frac{-\text{sech}(x^2) \tanh(x^2)}{1 + (\text{sech}(x^2))^2} $$
Steps to Solve
- Identify the function and variables
Let's define $u = x^2$. We need to differentiate $f(u) = \tan^{-1}(\text{sech}(u))$ with respect to $u$.
- Differentiate the outer function
Using the derivative of the inverse tangent function, we have: $$ \frac{df}{du} = \frac{1}{1 + (\text{sech}(u))^2} $$
- Differentiate the inner function
Next, differentiate the inner function $\text{sech}(u)$ using its derivative: $$ \text{sech}(u) = \frac{1}{\cosh(u)} $$ The derivative of $\text{sech}(u)$ is: $$ \frac{d}{du} \text{sech}(u) = -\text{sech}(u) \tanh(u) $$
- Apply the chain rule
Now we apply the chain rule: $$ \frac{df}{dx^2} = \frac{df}{du} \cdot \frac{du}{dx^2} $$ Since $u = x^2$, we have $\frac{du}{dx^2} = 1$. Hence, $$ \frac{df}{dx^2} = \frac{1}{1 + (\text{sech}(u))^2} \cdot (-\text{sech}(u) \tanh(u)) $$
- Combine the results
So the final result becomes: $$ \frac{d}{dx^2} \tan^{-1}(\text{sech}(x^2)) = \frac{-\text{sech}(x^2) \tanh(x^2)}{1 + (\text{sech}(x^2))^2} $$
$$ \frac{d}{dx^2} \tan^{-1}(\text{sech}(x^2)) = \frac{-\text{sech}(x^2) \tanh(x^2)}{1 + (\text{sech}(x^2))^2} $$
More Information
This derivative involves a combination of the inverse tangent function and the hyperbolic secant function, showing how chain rule applications can simplify differentiation processes.
Tips
- Confusing the derivative of $\tan^{-1}(x)$ with the function itself. Remember the correct derivative is $ \frac{1}{1 + x^2} $ and adjust for the inner function correctly.
- Neglecting to apply the chain rule properly when differentiating composite functions.
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