If f(x) = sin left(cosh left(frac{x^2+1}{x^2+ | vedclass

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(A) Given $f(x) = \sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$.Applying the chain rule, we differentiate with respect to $x$:$f^{\prime}(

Vedclass · Accepted Answer

$\frac{2}{9} \sinh \left(\frac{2}{3}\right) \cos \left(\cosh \left(\frac{2}{3}\right)\right)$

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(A) Given $f(x) = \sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$.Applying the chain rule, we differentiate with respect to $x$:$f^{\prime}(x) = \cos \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right) \cdot \sinh \left(\frac{x^2+1}{x^2+2}\right) \cdot \frac{d}{dx} \left(\frac{x^2+1}{x^2+2}\right)$.Using the quotient rule for the inner derivative:$\frac{d}{dx} \left(\frac{x^2+1}{x^2+2}\right) = \frac{(x^2+2)(2x) - (x^2+1)(2x)}{(x^2+2)^2} = \frac{2x^3 + 4x - 2x^3 - 2x}{(x^2+2)^2} = \frac{2x}{(x^2+2)^2}$.Thus, $f^{\prime}(x) = \cos \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right) \cdot \sinh \left(\frac{x^2+1}{x^2+2}\right) \cdot \frac{2x}{(x^2+2)^2}$.Evaluating at $x = 1$:$f^{\prime}(1) = \cos \left(\cosh \left(\frac{1^2+1}{1^2+2}\right)\right) \cdot \sinh \left(\frac{1^2+1}{1^2+2}\right) \cdot \frac{2(1)}{(1^2+2)^2} = \frac{2}{9} \sinh \left(\frac{2}{3}\right) \cos \left(\cosh \left(\frac{2}{3}\right)\right)$.

If $f(x) = \sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$, then $f^{\prime}(1) = $

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