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Differentiate the following functions: f(x) = x * tan2(1/x) and y = cos(sqrt(1 + sin2(x)))

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Here are the derivatives for each function:

  1. To differentiate f(x)=x⋅tan⁡2(1/x)f(x) = x \cdot \tan^2(1/x), we use the product rule f′(x)=u′v+uv′f'(x) = u'v + uv'. Let u=xu = x and v=tan⁡2(1/x)v = \tan^2(1/x).

  1. Find u′u'.

    $$
u' = \frac{d}{dx}(x) = 1
$$

  2. Find v′v' using the chain rule.

    $$
v = (\tan(x^{-1}))^2
$$
$$
v' = 2\tan(x^{-1}) \cdot \frac{d}{dx}(\tan(x^{-1}))
$$
$$
v' = 2\tan(x^{-1}) \cdot \sec^2(x^{-1}) \cdot \frac{d}{dx}(x^{-1})
$$
$$
v' = 2\tan(1/x) \cdot \sec^2(1/x) \cdot (-1x^{-2})
$$
$$
v' = -\frac{2}{x^2}\tan(1/x)\sec^2(1/x)
$$

  3. Apply the product rule.

    $$
f'(x) = 1 \cdot \tan^2(1/x) + x \cdot \left(-\frac{2}{x^2}\tan(1/x)\sec^2(1/x)\right)
$$
$$
f'(x) = \tan^2(1/x) - \frac{2}{x}\tan(1/x)\sec^2(1/x)
$$
The derivative is:
$$
\boxed{\tan^2(1/x) - \frac{2}{x}\tan(1/x)\sec^2(1/x)}
$$

    7. To differentiate y=cos⁡(1+sin⁡2(x))y = \cos(\sqrt{1 + \sin^2(x)}), we use the chain rule multiple times.

  4. Differentiate the outermost function, cos⁡(u)\cos(u).

    $$
\frac{dy}{dx} = -\sin(\sqrt{1 + \sin^2(x)}) \cdot \frac{d}{dx}(\sqrt{1 + \sin^2(x)})
$$

  5. Differentiate 1+sin⁡2(x)=(1+sin⁡2(x))1/2\sqrt{1 + \sin^2(x)} = (1 + \sin^2(x))^{1/2}.

    $$
\frac{d}{dx}((1 + \sin^2(x))^{1/2}) = \frac{1}{2}(1 + \sin^2(x))^{-1/2} \cdot \frac{d}{dx}(1 + \sin^2(x))
$$

  6. Differentiate 1+sin⁡2(x)1 + \sin^2(x).

    $$
\frac{d}{dx}(1 + \sin^2(x)) = 0 + 2\sin(x)\cos(x)
$$

  7. Combine the results.

    $$
\frac{dy}{dx} = -\sin(\sqrt{1 + \sin^2(x)}) \cdot \frac{1}{2\sqrt{1 + \sin^2(x)}} \cdot (2\sin(x)\cos(x))
$$
$$
\frac{dy}{dx} = -\frac{\sin(x)\cos(x)\sin(\sqrt{1 + \sin^2(x)})}{\sqrt{1 + \sin^2(x)}}
$$
The derivative is:
$$
\boxed{-\frac{\sin(x)\cos(x)\sin(\sqrt{1 + \sin^2(x)})}{\sqrt{1 + \sin^2(x)}}}
$$

    8. To differentiate y=ln⁡(xsin⁡(x)1−x2)y = \ln\left(\frac{x\sin(x)}{\sqrt{1-x^2}}\right), we first simplify using logarithm properties.

  8. Apply logarithm properties ln⁡(a/b)=ln⁡(a)−ln⁡(b)\ln(a/b) = \ln(a) - \ln(b) and ln⁡(ab)=ln⁡(a)+ln⁡(b)\ln(ab) = \ln(a) + \ln(b).

    $$
y = \ln(x\sin(x)) - \ln(\sqrt{1-x^2})
$$
$$
y = \ln(x) + \ln(\sin(x)) - \frac{1}{2}\ln(1-x^2)
$$

  9. Differentiate each term.

    $$
\frac{d}{dx}(\ln(x)) = \frac{1}{x}
$$
$$
\frac{d}{dx}(\ln(\sin(x))) = \frac{1}{\sin(x)} \cdot \cos(x) = \cot(x)
$$
$$
\frac{d}{dx}\left(-\frac{1}{2}\ln(1-x^2)\right) = -\frac{1}{2} \cdot \frac{1}{1-x^2} \cdot (-2x) = \frac{x}{1-x^2}
$$

  10. Combine the derivatives.

    $$
\frac{dy}{dx} = \frac{1}{x} + \cot(x) + \frac{x}{1-x^2}
$$
The derivative is:
$$
\boxed{\frac{1}{x} + \cot(x) + \frac{x}{1-x^2}}
$$

    9. To differentiate y=sec⁡2(sin⁡(x))y = \sec^2(\sqrt{\sin(x)}), we use the chain rule multiple times.

  11. Rewrite the function.

    $$
y = (\sec((\sin(x))^{1/2}))^2
$$

  12. Apply the power rule and chain rule.

    $$
\frac{dy}{dx} = 2\sec(\sqrt{\sin(x)}) \cdot \frac{d}{dx}(\sec(\sqrt{\sin(x)}))
$$

  13. Differentiate sec⁡(sin⁡(x))\sec(\sqrt{\sin(x)}).

    $$
\frac{d}{dx}(\sec(\sqrt{\sin(x)})) = \sec(\sqrt{\sin(x)})\tan(\sqrt{\sin(x)}) \cdot \frac{d}{dx}(\sqrt{\sin(x)})
$$

  14. Differentiate sin⁡(x)=(sin⁡(x))1/2\sqrt{\sin(x)} = (\sin(x))^{1/2}.

    $$
\frac{d}{dx}((\sin(x))^{1/2}) = \frac{1}{2}(\sin(x))^{-1/2} \cdot \frac{d}{dx}(\sin(x))
$$
$$
\frac{d}{dx}((\sin(x))^{1/2}) = \frac{1}{2\sqrt{\sin(x)}} \cdot \cos(x) = \frac{\cos(x)}{2\sqrt{\sin(x)}}
$$

  15. Combine all parts.

    $$
\frac{dy}{dx} = 2\sec(\sqrt{\sin(x)}) \cdot \sec(\sqrt{\sin(x)})\tan(\sqrt{\sin(x)}) \cdot \frac{\cos(x)}{2\sqrt{\sin(x)}}
$$
$$
\frac{dy}{dx} = \frac{2\sec^2(\sqrt{\sin(x)})\tan(\sqrt{\sin(x)})\cos(x)}{2\sqrt{\sin(x)}}
$$
$$
\frac{dy}{dx} = \frac{\sec^2(\sqrt{\sin(x)})\tan(\sqrt{\sin(x)})\cos(x)}{\sqrt{\sin(x)}}
$$
The derivative is:
$$
\boxed{\frac{\sec^2(\sqrt{\sin(x)})\tan(\sqrt{\sin(x)})\cos(x)}{\sqrt{\sin(x)}}}
$$

    10. To differentiate y=ln⁡(cos⁡(5x2+3))y = \ln(\sqrt{\cos(5x^2 + 3)}), we first simplify using logarithm properties.

  16. Apply logarithm property ln⁡(ab)=bln⁡(a)\ln(a^b) = b\ln(a).

    $$
y = \ln((\cos(5x^2 + 3))^{1/2})
$$
$$
y = \frac{1}{2}\ln(\cos(5x^2 + 3))
$$

  17. Differentiate using the chain rule.

    $$
\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\cos(5x^2 + 3)} \cdot \frac{d}{dx}(\cos(5x^2 + 3))
$$

  18. Differentiate cos⁡(5x2+3)\cos(5x^2 + 3).

    $$
\frac{d}{dx}(\cos(5x^2 + 3)) = -\sin(5x^2 + 3) \cdot \frac{d}{dx}(5x^2 + 3)
$$

  19. Differentiate 5x2+35x^2 + 3.

    $$
\frac{d}{dx}(5x^2 + 3) = 10x
$$

  20. Combine the results.

    $$
\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\cos(5x^2 + 3)} \cdot (-\sin(5x^2 + 3)) \cdot (10x)
$$
$$
\frac{dy}{dx} = -\frac{10x\sin(5x^2 + 3)}{2\cos(5x^2 + 3)}
$$
$$
\frac{dy}{dx} = -5x\tan(5x^2 + 3)
$$
The derivative is:
$$
\boxed{-5x\tan(5x^2 + 3)}
$$
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