Question
Download Solution PDF\(\rm \frac{d}{dx}x^{2x}\) =
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा:
\(\rm log\ x^n=nlog\ x\)
\(\rm \frac{d}{dx}[log \ x ]=\frac1x\)
गुणनफल नियम:
\(\rm \frac{d}{dx}[f(x)g(x)]=\frac{d}{dx}[f(x)] \ g(x) + \frac{d}{dx}[g(x)]\ f(x)\)
गणना:
माना कि y = x2x
दोनों पक्षों पर लॉग लेते हुए हम प्राप्त करते हैं
log y = log( x2x ) = 2x logx
(∵ \(\rm log\ x^n=nlog\ x\))
अब, अवकलज लेते हुए,
\(\rm \frac{d}{dx}[log \ y]\) = 2{ \(\rm \frac{d}{dx}[x] \ log \ x + \frac{d}{dx}[log \ x]\ x\) }
\(\rm \frac1y\frac{dy}{dx}\) = 2[log x + x. \(\rm \frac1x\)]
(∵ \(\rm \frac{d}{dx}[log \ x ]=\frac1x\))
⇒ \(\rm \frac{dy}{dx}\) = 2y [1 + log x]
⇒ \(\rm \frac{dy}{dx}\) = 2x2x [1 + log x]
(∵ y = x2x )
इसलिए, विकल्प (4) सही है।
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