Differentiate -x-1x-2-x-3x-4x-5 with respect to x-

The correct option is C 1/2√((x 1)(x 2)/(x 3)(x 4)(x 5))[1/x 1+1/x 2 1/x 3 1/x 4 1/x 5] Let y=√((x 1)(x 2)/(x 3)(x 4)(x 5)) Taking logarithm on both ...

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Byju's Answer

Standard XII

Mathematics

Logarithmic Differentiation

Differentiate...

Question

Differentiate $\sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}}$ with respect to $x$ .

\frac{1}{2} \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}} [\frac{1}{x - 1} - \frac{1}{x - 2} - \frac{1}{x - 3} - \frac{1}{x - 4} - \frac{1}{x - 5}]

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\frac{1}{2} \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}} [\frac{1}{x - 2} + \frac{1}{x - 3} + \frac{1}{x - 4} + \frac{1}{x - 5}]

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\frac{1}{2} \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}} [\frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{x - 3} - \frac{1}{x - 4} - \frac{1}{x - 5}]

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Solution

The correct option is C $\frac{1}{2} \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}} [\frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{x - 3} - \frac{1}{x - 4} - \frac{1}{x - 5}]$
Let $y = \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}}$
Taking logarithm on both sides, we get
$log y = l o g \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}}$
$= \frac{1}{2} [log (x - 1) + log (x - 2) - log (x - 3) - log (x - 4) - log (x - 5)]$ ....................(Since $log \frac{a}{b} = log a - log b$ and $log a b = log a + log b$ )
Differentiating both sides w.r.t. $x$ , we get
$\frac{1}{y} \frac{d y}{d x} = \frac{1}{2} (\frac{1}{x - 1} \frac{d}{d x} (x - 1) + \frac{1}{x - 2} \frac{d}{d x} (x - 2) - \frac{1}{x - 3} \frac{d}{d x} (x - 3) - \frac{1}{x - 4} \frac{d}{d x} (x - 4) - \frac{1}{x - 5} \frac{d}{d x} (x - 5))$
$∴ \frac{d y}{d x} = \frac{1}{2} \sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}} [\frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{x - 3} - \frac{1}{x - 4} - \frac{1}{x - 5}]$

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Similar questions

Q. Solve for x:

(i)

\frac{x - 1}{x - 2} + \frac{x - 3}{x - 4} = 3 \frac{1}{3}; x \neq 2, 4

(ii)

\frac{1}{x} - \frac{1}{x - 2} = 3, x \neq 0, 2

(iii)

x + \frac{1}{x} = 3, x \neq 0

(iv)

\frac{16}{x} - 1 = \frac{15}{x + 1}, x \neq 0, - 1

(v)

\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5

$\int \frac{5}{x^{4} + 1} d x$

Solve :

$(i) \frac{1}{3} x - 6 = \frac{5}{2} (i i) \frac{2 x}{3} - \frac{3 x}{8} = \frac{7}{12} (i i i) (x + 2) (x + 3) + (x - 3) (x - 2) - 2 x (x + 1) = 0 (i v) \frac{1}{10} - \frac{7}{x} = 35 (v) 13 (x - 4) - 3 (x - 9) - 4 (x + 4) = 0 (v i) x + 7 - \frac{8 x}{3} = \frac{17 x}{6} - \frac{5 x}{8} (v i i) \frac{3 x - 2}{4} - \frac{2 x + 3}{3} = \frac{2}{3} - x (v i i i) \frac{x + 2}{6} - (\frac{11 - x}{3} - \frac{1}{4}) = \frac{3 x - 4}{12} (i x) \frac{2}{5 x} - \frac{5}{3 x} = \frac{1}{15} (x) \frac{x + 2}{3} - \frac{x + 1}{5} = \frac{x - 3}{4} - 1 (x i) \frac{3 x - 2}{3} + \frac{2 x + 3}{2} = x + \frac{7}{6} (x i i) x - \frac{x - 1}{2} = 1 - \frac{x - 2}{3} (x i i i) \frac{9 x + 7}{2} - (x - \frac{x - 2}{7}) = 36 (x i v) \frac{6 x + 1}{2} + 1 = \frac{7 x - 3}{3}$

Find the derivative of

(i) (ii) (5x³ + 3x – 1) (x – 1)

(iii) x^–3 (5 + 3x) (iv) x⁵ (3 – 6x^–9)

(v) x^–4 (3 – 4x^–5) (vi)

\frac{1}{(x - 1) (x - 2)} + \frac{1}{(x - 2) (x - 3)} + \frac{1}{(x - 3) (x - 4)} = \frac{1}{6}

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Differentiate √(x−1)(x−2)(x−3)(x−4)(x−5) with respect to x.

Differentiate $\sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4) (x - 5)}}$ with respect to $x$ .