A) \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+x)\log x \right]\]
B) \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-x)\log x \right]\]
C) \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+2x)\log x \right]\]
D) \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-2x)\log x \right]\]
Correct Answer: C
Solution :
\[y=\log x.{{e}^{(\tan x+{{x}^{2)}}}}\] \[\therefore \frac{dy}{dx}={{e}^{(\tan x+{{x}^{2}})}}.\frac{1}{x}+\log x.{{e}^{(\tan x+{{x}^{2}})}}({{\sec }^{2}}x+2x)\] \[={{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+2x)\log x \right]\].
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