Question

If y=logx³ then d²y/dx² = __________​

Answer 1

Explanation:

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Answer 2

Answer:

$- \frac{3} { {x}^{2} ln(10) }$

Explanation:

Differential principle for logarithm:

$y = log_{a}(x) \\ \frac{dy}{dx} = \frac{1}{x ln(a) }$

If

y = log x³

y = 3 log x

$y = 3 log_{10}(x)$

Product differentiation formula:

y = UV

$\frac{dy}{dx} = (u \times \frac{dv}{dx}) + (v \times \frac{du}{dx} )$

Take,

$u = 3 \\ \frac{du}{dx} = 0$

$v = log_{10}(x) \\ \frac{dv}{dx} = \frac{1}{x ln(10) }$

$\frac{dy}{dx} = (3 \times \frac{1}{x ln(10) } ) + 0 \\ = \frac{3}{x ln(10) } \\ = \frac{3 {x}^{ - 1} }{ ln(10) }$

To get d²y/dx²:

Take,

$u = \frac{3}{ ln(10) } \\ \frac{du}{dx} = 0$

$v = {x}^{ - 1} \\ \frac{dv}{dx} = - {x}^{ - 2} \\ = - \frac{1}{ {x}^{2} }$

Finally,

$\frac{ {d}^{2}y}{d {x}^{2} } = (\frac{3} { ln(10) } \times - \frac{1}{ {x}^{2} } ) + 0 \\ = - \frac{3} { {x}^{2} ln(10) }$