Let y=d/dx (x^3log_ex) then using multilication law y = x^3 d/dx(log_e^x) + log_ex d/dx (x^3) y =x^3/x+ log_ex · 2x^2 y= x^2 +2x^2 log_ex ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Derivatives

Find ddx x3...

Question

Find $\frac{d}{d x} (x^{3} {log}_{e} x)$

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Solution

Let $y = \frac{d}{d x} (x^{3} l o g_{e} x)$
then using multilication law
$y = x^{3} \frac{d}{d x} (l o g_{e^{x}}) + l o g_{e} x \frac{d}{d x} (x^{3})$
$y = \frac{x^{3}}{x} + l o g_{e} x \cdot 2 x^{2}$
$y = x^{2} + 2 x^{2} l o g_{e} x$
$y = x^{2} [1 + l o g_{e} x^{2}]$

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Find ddx(x3logex)

Let y=ddx(x3logex)then using multilication lawy=x3ddx(logex)+logexddx(x3)y=x3x+logex⋅2x2y=x2+2x2logexy=x2[1+logex2]

Find $\frac{d}{d x} (x^{3} {log}_{e} x)$

Let $y = \frac{d}{d x} (x^{3} l o g_{e} x)$
then using multilication law
$y = x^{3} \frac{d}{d x} (l o g_{e^{x}}) + l o g_{e} x \frac{d}{d x} (x^{3})$
$y = \frac{x^{3}}{x} + l o g_{e} x \cdot 2 x^{2}$
$y = x^{2} + 2 x^{2} l o g_{e} x$
$y = x^{2} [1 + l o g_{e} x^{2}]$