Question

differentiate 6x-1/x^2-2x​

Answer 1

Answer:-

We have to find:-

$\sf \dfrac{d}{dx} \bigg( \dfrac{6x - 1}{ {x}^{2} - 2x } \bigg)$

Using d/dx (u/v) = (v * du/dx - u * dv/dx) / v² we get,

$\implies \sf \: \dfrac{( {x}^{2} - 2x) \big( \frac{d}{dx} (6x - 1) \big) - (6x - 1) \big( \frac{d}{dx}( {x}^{2} - 2x) \big)}{ {( {x}^{2} - 2x) }^{2} }$

Using d/dx (u ± v) = d/dx (u) ± d/dx (v) we get,

$\implies \sf \: \dfrac{( {x}^{2} - 2x) \big( \frac{d}{dx} (6x) - \frac{d}{dx} (1) \big) - (6x - 1) \big( \frac{d}{dx} ( {x}^{2}) - \frac{d}{dx} (2x) \big)}{ {( {x}^{2} - 2x)}^{2} }$

Using d/dx (xⁿ) = n × xⁿ⁻¹ & d/dx (constant) = 0 we get,

$\implies \sf \: \dfrac{( {x}^{2} - 2x)(6 \times 1 \times {x}^{1 - 1} - 0) - (6x - 1)(2 \times {x}^{2 - 1} - 2 \times 1 \times {x}^{1-1} }{ { ({x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \: \dfrac{( {x}^{2} - 2x)(6) - (6x - 1)(2x-2) }{ {( {x}^{2} - 2x) }^{2} } \\ \\ \\ \implies \sf \: \dfrac{6 {x}^{2} - 12x - [6x(2x-2) -1(2x-2)]}{( { {x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \: \frac{6 {x}^{2} - 12x - (12x^2 - 12x - 2x + 2)}{ {( {x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \dfrac{6x^2 - 12x - 12x^2 + 12x + 2x - 2}{(x^2-2x)^2} \\ \\ \\ \implies \sf \dfrac{-6x^2+2x-2}{(x^2 - 2x)^2}$

Answer 2

$\huge\bf\fbox\red{Answer:-}$

Using d/dx (u/v) = (v * du/dx - u * dv/dx) / v² we get,

$\implies \sf \: \dfrac{( {x}^{2} - 2x) \big( \frac{d}{dx} (6x - 1) \big) - (6x - 1) \big( \frac{d}{dx}( {x}^{2} - 2x) \big)}{ {( {x}^{2} - 2x) }^{2} }$

Using d/dx (xⁿ) = n × xⁿ⁻¹ & d/dx (constant) = 0 we get,

$\begin{gathered} \implies \sf \: \dfrac{( {x}^{2} - 2x)(6 \times 1 \times {x}^{1 - 1} - 0) - (6x - 1)(2 \times {x}^{2 - 1} - 2 \times 1 \times {x}^{1-1} }{ { ({x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \: \dfrac{( {x}^{2} - 2x)(6) - (6x - 1)(2x-2) }{ {( {x}^{2} - 2x) }^{2} } \\ \\ \\ \implies \sf \: \dfrac{6 {x}^{2} - 12x - [6x(2x-2) -1(2x-2)]}{( { {x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \: \frac{6 {x}^{2} - 12x - (12x^2 - 12x - 2x + 2)}{ {( {x}^{2} - 2x)}^{2} } \\ \\ \\ \implies \sf \dfrac{6x^2 - 12x - 12x^2 + 12x + 2x - 2}{(x^2-2x)^2} \\ \\ \\ \implies \sf \dfrac{-6x^2+2x-2}{(x^2 - 2x)^2} \end{gathered}$