Differentiate w.r.t. x the function: (5x)3cos2x - Mathematics

प्रश्न

Differentiate w.r.t. x the function:

${(5 x)}^{3 \cos 2 x}$

योग

उत्तर

Let y = ${(5 x)}^{3 \cos 2 x}$

Taking log on both sides, we get

log y = 3 cos 2x log (5x) = 3 cos 2x [log 5 + log x]

log y = 3 cos 2x log 5 + 3 cos 2x log x ....(1)

Differentiating (1) w.r.t.x, we get

$\frac{1}{y} \frac{d y}{d x} = 3 \log 5 (- \sin 2 x) \cdot 2 + \frac{3 \cos 2 x}{x} + 3 \log x (- 2 \sin 2 x)$

$= - 6 \log 5 \sin 2 x + \frac{3 \cos 2 x}{x} - 6 \log x \sin 2 x$

$\frac{d y}{d x} = {(5 x)}^{3 \cos 2 x} [\frac{3 \cos 2 x}{x} - 6 (\log 5 + \log x) \sin 2 x]$

$= {(5 x)}^{3 \cos 2 x} [\frac{3 \cos 2 x}{x} - 6 \log 5 x \sin 2 x]$

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Concept of Differentiability

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अध्याय 5: Continuity and Differentiability - Exercise 5.9 [पृष्ठ १९१]

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अध्याय 5 Continuity and Differentiability
Exercise 5.9 | Q 3 | पृष्ठ १९१

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Differentiate w.r.t. x the function: (5x)3cos2x - Mathematics

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Differentiate w.r.t. x the function: (5x)3cos2x - Mathematics

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