find the 3^(rd) derivative of x^5 ln(2x) using the Leibniz theorem


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Question Number 83189 by john santu last updated on 28/Feb/20

$find the 3^{rd} derivative of$ $x^{5} \ln (2 x) using the Leibniz theorem$
Commented by jagoll last updated on 28/Feb/20

$y = x^{5} \ln (2 x)$ $y^{'} (x) = {5 x}^{4} \ln (2 x) + \frac{{2 x}^{5}}{2 x}$ $y^{'} (x) = {5 x}^{4} \ln (2 x) + x^{4}$ $y^{″} (x) = {20 x}^{3} \ln (2 x) + \frac{{10 x}^{4}}{2 x} + {4 x}^{3}$ $y^{″} (x) = {20 x}^{3} \ln (2 x) + {9 x}^{3}$ $y^{‴} (x) = {60 x}^{2} \ln (2 x) + \frac{{40 x}^{3}}{2 x} + {27 x}^{2}$ $y^{‴} (x) = {60 x}^{2} \ln (2 x) + {47 x}^{2}$
Commented by mr W last updated on 28/Feb/20

${(x^{5} \ln (2 x))}^{(3)}$ $= x^{5} (\frac{2}{x^{3}}) + 3 (5 x^{4}) (- \frac{1}{x^{2}}) + 3 (20 x^{3}) (\frac{1}{x}) + (60 x^{2}) \ln (2 x)$ $= 2 x^{2} - 15 x^{2} + 60 x^{2} + 60 x^{2} \ln (2 x)$ $= x^{2} (47 + 60 \ln (2 x))$
Commented by mathmax by abdo last updated on 28/Feb/20
$direct calculus f(x)=x^5 ln(2x) ⇒ f^((1)) (x)=5x^4 ln(2x) +x^5 ×(1/x) =5x^4 ln(2x)+x^4 f^((2)) (x)=20x^3 ln(2x)+5x^4 ×(1/x) +4x^3 =20x^3 ln(2x)+5x^3 +4x^3 =20x^3 ln(2x)+9x^3 ⇒f^((3)) (x)=60x^2 ln(2x)+20x^3 ×(1/x) +27x^2 =60x^2 ln(2x) +47 x^2 =x^2 {60ln(2x)+47}$
$d i r e c t c a l c u l u s f (x) = x^{5} l n (2 x) \Rightarrow$ $f^{(1)} (x) = 5 x^{4} l n (2 x) + x^{5} \times \frac{1}{x} = 5 x^{4} l n (2 x) + x^{4}$ $f^{(2)} (x) = 20 x^{3} l n (2 x) + 5 x^{4} \times \frac{1}{x} + 4 x^{3} = 20 x^{3} l n (2 x) + 5 x^{3} + 4 x^{3}$ $= 20 x^{3} l n (2 x) + 9 x^{3} \Rightarrow f^{(3)} (x) = 60 x^{2} l n (2 x) + 20 x^{3} \times \frac{1}{x} + 27 x^{2}$ $= 60 x^{2} l n (2 x) + 47 x^{2} = x^{2} {60 l n (2 x) + 47}$
Commented by jagoll last updated on 28/Feb/20

$what generally Leibniz theorem sir$
Commented by jagoll last updated on 28/Feb/20

$yes sir . thank you$
Commented by mathmax by abdo last updated on 29/Feb/20

$i f t h e f u n c t i o n s f a n d g a r e C^{n} o n I \subset R (o r C) w e h a v e$ ${(f . g)}^{(n)} = \sum_{k = 0}^{n} C_{n}^{k} f^{(k)} g^{(n - k)} (l e i b n i z f o r m u l a f o r d e r i v a t i o n)$
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