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differentiate with respect to x e^sin x + (tan x)^x

differentiate with respect to x
e^sin x + (tan x)^x

Grade:10

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
6 years ago

Hello Student,
\\ \\ \frac{d}{dx}\left(e^{\sin \left(x\right)}+\tan ^x\left(x\right)\right) \\ =\frac{d}{dx}\left(e^{\sin \left(x\right)}\right)+\frac{d}{dx}\left(\tan ^x\left(x\right)\right) \\ \frac{d}{dx}\left(e^{\sin \left(x\right)}\right) \\ =\frac{d}{du}\left(e^u\right)\frac{d}{dx}\left(\sin \left(x\right)\right) \\ \frac{d}{dx}\left(\sin \left(x\right)\right) \\ =\cos \left(x\right) \\ =e^{\sin \left(x\right)}\cos \left(x\right) \\ \frac{d}{dx}\left(\tan ^x\left(x\right)\right)
\\ =\frac{d}{du}\left(e^u\right)\frac{d}{dx}\left(x\ln \left(\tan \left(x\right)\right)\right) \\ \frac{d}{dx}\left(x\ln \left(\tan \left(x\right)\right)\right) \\ =\frac{d}{dx}\left(x\right)\ln \left(\tan \left(x\right)\right)+\frac{d}{dx}\left(\ln \left(\tan \left(x\right)\right)\right)x \\ =\frac{x}{\cos ^2\left(x\right)\tan \left(x\right)}+\ln \left(\tan \left(x\right)\right) \\ =e^{x\ln \left(\tan \left(x\right)\right)}\left(\frac{x}{\cos ^2\left(x\right)\tan \left(x\right)}+\ln \left(\tan \left(x\right)\right)\right) \\ =e^{\sin \left(x\right)}\cos \left(x\right)+\tan ^x\left(x\right)\left(\frac{x}{\cos ^2\left(x\right)\tan \left(x\right)}+\ln \left(\tan \left(x\right)\right)\right) \\

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty

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