Differentiate with respect to x : exsin x + xncos x

1.	Differentiate with respect to x : exsin x + xncos x
Answer» f(x) = e^x sin x +xⁿ cos x ∴ f'(x) = \(\frac{d}{dx}\){e^x sin x + xⁿ cos x} = \(\frac{d}{dx}\)(e^xsin x)+\(\frac{d}{dx}\)(xⁿcos x) = sinx\(\frac{d}{dx}\)e^x + e^x\(\frac{d}{dx}\)sinx + cosx\(\frac{d}{dx}\)xⁿ + xⁿ\(\frac{d}{dx}\)cosx = sin.e^x + e^x . cos x + cos x ∙ nxⁿ⁻¹ + xⁿ . (− sin x) = e^x (sin x + cos x) + xⁿ⁻¹ [ncos x − xsin x]

Answer»

f(x) = e^x sin x +xⁿ cos x

∴ f'(x) = \(\frac{d}{dx}\){e^x sin x + xⁿ cos x}

= \(\frac{d}{dx}\)(e^xsin x)+\(\frac{d}{dx}\)(xⁿcos x)

= sinx\(\frac{d}{dx}\)e^x + e^x\(\frac{d}{dx}\)sinx + cosx\(\frac{d}{dx}\)xⁿ + xⁿ\(\frac{d}{dx}\)cosx

= sin.e^x + e^x . cos x + cos x ∙ nxⁿ⁻¹ + xⁿ . (− sin x)

= e^x (sin x + cos x) + xⁿ⁻¹ [ncos x − xsin x]

Discussion