Find the derivative of-i 2x- 34ii 5x3 - 3x- 1 x-1iii x-3 5 - 3xiv x5 3 - 6x-9v x-4 3 - 4x-5vi 2x- 1 - x23x- 1

(i) nbsp; Let f (x) = 2x 34 Differentiation with respect to x ⇒ f' (x) = ddx nbsp; (2x 34 ) ⇒ f' (x) = 2.1 0 ∴ f' (x) = 2 (ii) nbsp; Let f(x) = (5x^3 ...

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Byju's Answer

Standard XII

Mathematics

Product Rule of Differentiation

Find the deri...

Question

Find the derivative of:
$(i)$ $2 x - \frac{3}{4}$
$(i i)$ $(5 x^{3} + 3 x - 1) (x - 1)$
$(i i i)$ $x^{- 3} (5 + 3 x)$
$(i v)$ $(x^{5}) (3 - 6 x^{- 9})$
$(v)$ $(x^{- 4}) (3 - 4 x^{- 5})$
$(v i)$ $\frac{2}{x + 1} - \frac{x^{2}}{3 x - 1}$

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Solution

$(i)$ Let $f (x) = 2 x - \frac{3}{4}$
Differentiation with respect to $x$
$\Rightarrow f^{'} (x) = \frac{d}{d x} (2 x - \frac{3}{4})$
$\Rightarrow f^{'} (x) = 2.1 - 0$
$∴ f^{'} (x) = 2$

$(i i)$ Let $f (x) = (5 x^{3} + 3 x - 1) (x - 1)$
Differentiation with respect to $x$
$\Rightarrow f^{'} (x) = \frac{d}{d x} ((5 x^{3} + 3 x - 1) (x - 1))$
Using $\frac{d}{d x} (u v) = u^{'} v + u v^{'}$
$f^{'} (x) = (\frac{d}{d x} (5 x^{3} + 3 x - 1)) (x - 1) + (\frac{d}{d x} (x - 1)) (5 x^{3} + 3 x - 1)$
$\Rightarrow f^{'} (x) = (15 x^{2} + 3) (x - 1) + (1) (5 x^{3} + 3 x - 1)$
$\Rightarrow f^{'} (x) = 15 x^{2} (x - 1) + 3 (x - 1) + 5 x^{3} + 3 x - 1$
$\Rightarrow f^{'} (x) = 15 x^{3} - 15 x^{2} + 3 x - 3 + 5 x^{3} + 3 x - 1$
$\Rightarrow f^{'} (x) = 15 x^{3} + 5 x^{3} - 15 x^{2} + 3 x + 3 x - 3 - 1$
$\Rightarrow f^{'} (x) = 20 x^{3} - 15 x^{2} + 6 x - 4$

$(i i i)$ Let $f (x) = x^{- 3} (5 + 3 x)$
Differentiating with respect to $x$
$\Rightarrow f^{'} (x) = \frac{d}{d x} (x^{- 3} (5 + 3 x))$
$\Rightarrow f^{'} (x) = (\frac{d}{d x} (x^{- 3})) (5 + 3 x) + (\frac{d}{d x} (5 + 3 x)) (x^{- 3})$
$\Rightarrow f^{'} (x) = - 3 x^{- 4} (5 + 3 x) + 3 (x^{- 3})$
$\Rightarrow f^{'} (x) = - 15 x^{- 4} - 9 x^{- 4 + 1} + 3 x^{- 3}$
$\Rightarrow f^{'} (x) = - 15 x^{- 4} - 9 x^{- 3} + 3 x^{- 3}$
$\Rightarrow f^{'} (x) = - 15 x^{- 4} - 6 x^{- 3}$
$\Rightarrow f^{'} (x) = \frac{- 15}{x^{4}} - \frac{6}{x^{3}}$
$\Rightarrow f^{'} (x) = - 3 [\frac{5}{x^{4}} + \frac{2}{x^{3}}]$
$\Rightarrow f^{'} (x) = - 3 [\frac{5 + 2 x}{x^{4}} +]$
$\Rightarrow f^{'} (x) = \frac{- 3}{x^{4}} (5 + 2 x)$

$(i v)$ Let $f (x) = x^{5} (3 - 6 x^{- 9})$
Differentiating with respect to $x$
$\Rightarrow f^{'} (x) = \frac{d}{d x} (x^{5} (3 - 6 x^{- 9})$
$\Rightarrow f^{'} (x) = (\frac{d}{d x} (x^{5})) (3 - 6 x^{- 9}) + (\frac{d}{d x} (3 - 6 x^{- 9})) (x^{5})$
$\Rightarrow f^{'} (x) == 5 x^{4} (3 - 6 x^{- 9}) + 54 x^{- 10} (x^{5})$
$\Rightarrow f^{'} (x) = 15 x^{4} - 30 x^{- 9 + 4} + 54 x^{- 10 + 5}$
$\Rightarrow f^{'} (x) == 15 x^{4} - 30 x^{- 5} + 54 x^{- 5}$
$\Rightarrow f^{'} (x) == 15 x^{4} + 24 x^{- 5}$
$\Rightarrow f^{'} (x) == 15 x^{4} + \frac{24}{x^{5}}$

$(v)$ Let $f (x) = x^{- 4} (3 - 4 x^{- 5})$
$\Rightarrow f^{'} (x) = 3 x^{- 4} - 4 x^{- 5} \times x^{- 4}$
$\Rightarrow f^{'} (x) = 3 x^{- 4} - 4 x^{- 5 - 4}$
$\Rightarrow f^{'} (x) = 3 x^{- 4} - 4 x^{- 9}$
Differentiation with respect to $x$
$\Rightarrow f^{'} (x) = (3 x^{- 4} - 4 x^{- 9})^{'}$
$\Rightarrow f^{'} (x) = 3 (- 4 x^{- 4 - 1}) - 4 (- 9 x^{- 9 - 1})$
$\Rightarrow f^{'} (x) = 3 (- 4 x^{- 5}) - 4 (- 9 x^{- 10})$
$\Rightarrow f^{'} (x) = - 12 x^{- 5} + 36 x^{- 10}$
$\Rightarrow f^{'} (x) = \frac{- 12}{x^{5}} + \frac{36}{x^{10}}$

$(v i)$ Let $f (x) = \frac{2}{x + 1} - \frac{x^{2}}{3 x - 1}$
Differentitating with respect to $x$
$\Rightarrow f^{'} (x) = \frac{d}{d x} (\frac{2}{x + 1} - \frac{x^{2}}{3 x - 1})$
$\Rightarrow f^{'} (x) = \frac{d}{d x} (\frac{2}{x + 1}) - \frac{d}{d x} (\frac{x^{2}}{3 x - 1})$
$\Rightarrow f^{'} (x) = \frac{(x + 1) (2)^{'} - 2 (x + 1)^{'}}{(x + 1)^{2}} - \frac{(3 x - 1) (x^{2})^{'} - (3 x - 1)^{'} (x^{2})}{(3 x - 1)^{2}}$
$\Rightarrow f^{'} (x) = \frac{0 - 2 (1)}{(x + 1)^{2}} - \frac{2 x (3 x - 1) - 3 (x^{2})}{(3 x - 1)^{2}}$
$\Rightarrow f^{'} (x) = \frac{- 2}{(x + 1)^{2}} - \frac{6 x^{2} - 2 x - 3 x^{2}}{(3 x - 1)^{2}}$
$\Rightarrow f^{'} (x) = \frac{- 2}{(x + 1)^{2}} - \frac{x (3 x - 2)}{(3 x - 1)^{2}}$

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Similar questions

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)

Find the derivative of:

(i) 2x - $\frac{3}{4}$
(ii) $(5 x^{3} + 3 x - 1)$ (x-1)

(iii) $x^{- 3}$ (5+3x)
(iv) $x^{5} (3 - 6 x^{- 9})$

(v) $x^{- 4} (3 - 4 x^{- 5})$
(vi) $\frac{2}{x + 1} - \frac{x^{2}}{3 x - 1}$

Question 24
Factorise
(i) $2 x^{3} - 3 x^{2} - 17 x + 30$

(ii) $x^{3} - 6 x^{2} + 11 x - 6$

(iii) $x^{3} + x^{2} - 4 x - 4$

(iv) $3 x^{3} - x^{2} - 3 x + 1$

Q. Which of the following are quadratic equations?

(i) x² + 6x − 4 = 0
(ii)

\sqrt{3} x^{2} - 2 x + \frac{1}{2} = 0

(iii)

x^{2} + \frac{1}{x^{2}} = 5

(iv)

x - \frac{3}{x} = x^{2}

(v)

2 x^{2} - \sqrt{3 x} + 9 = 0

(vi)

x^{2} - 2 x - \sqrt{x} - 5 = 0

(vii) 3x² − 5x + 9 = x² − 7x + 3
(viii)

x + \frac{1}{x} = 1

(ix) x² − 3x = 0
(x)

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

(xi) (2x + 1) (3x + 2) = 6(x − 1) (x − 2)
(xii)

x + \frac{1}{x} = x^{2}, x \neq 0

(xiii) 16x² − 3 = (2x + 5) (5x − 3)
(xiv) (x + 2)³ = x³ − 4
(xv) x(x + 1) + 8 = (x + 2) (x − 2)

Check whether the following are quadratic equations :
(i) ${(x + 1)}^{2} = 2 (x - 3)$
(ii) $x^{2} - 2 x = (- 2) (3 - x)$
(iii) $(x - 2) (x + 1) = (x - 1) (x + 3)$
(iv) $(x - 3) (2 x + 1) = x (x + 5)$
(v) $(2 x - 1) (x - 3) = (x + 5) (x - 1)$
(vi) $x^{2} + 3 x + 1 = {(x - 2)}^{2}$
(vii) ${(x + 2)}^{3} = 2 x (x^{2} - 1)$
(viii) $x^{3} - 4 x^{2} - x + 1 = {(x - 2)}^{3}$

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Product Rule of Differentiation

Standard XII Mathematics

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Find the derivative of: (i) 2x−34 (ii) (5x3+3x−1)(x−1) (iii) x−3(5+3x) (iv) (x5)(3−6x−9) (v) (x−4)(3−4x−5) (vi) 2x+1−x23x−1

Find the derivative of:
$(i)$ $2 x - \frac{3}{4}$
$(i i)$ $(5 x^{3} + 3 x - 1) (x - 1)$
$(i i i)$ $x^{- 3} (5 + 3 x)$
$(i v)$ $(x^{5}) (3 - 6 x^{- 9})$
$(v)$ $(x^{- 4}) (3 - 4 x^{- 5})$
$(v i)$ $\frac{2}{x + 1} - \frac{x^{2}}{3 x - 1}$