let f,g,h are differentiable functions If f(0 )=1; g(0)=2; h(0)= 3 and the derivatives of their pair wise product - Maths - Limits and Derivatives

Question

let f,g,h are differentiable functions If f(0 )=1; g(0)=2;
h(0)= 3 and the derivatives of their pair wise product at x=0
are-

(fg)' (0)=6; (gh)'(0)=4; (hf)'(0)=5

then compute the value of (fgh)'(0)

Shivam Mishra · Accepted Answer

Hi Shubhangi,
Please find below the solution to the asked query:

Consider fxgxhx denoted as fghx
 Now if we differentiate fghx with respect to xwith the help of product rule we get,fgh'x=f'x
gxhx+g'x fxhx+h'x fxgxPut x=0⇒fgh'0=f0' g0h0+g'0 f0h0+h'0
f0g0Given thatf0=1 , g0=2 and h0=3⇒fgh'0=f0'
2 3+g'0 1 3+h'0 1
2⇒fgh'0=6f'0+3g'0+2h'0 ; equationiNow we need values of  f'0, g'0 and h'0
fg'x=f'x gx+g'x fx⇒fg'0=f'0 g0+g'0
f0But given that fg'0=6⇒6=f'0 2+g'0
1⇒2f'0+g'0=6  ; equationiigh'x=g'x hx+h'x
gx⇒gh'0=g'0 h0+h'0
g0But given that gh'0=4⇒4=g'0 3+h'0
2⇒3g'0+2h'0=4 ;equationiiifh'x=f'x hx++h'x
fx⇒fh'0=f'0 h0++h'0
f0But given that fh'0=5⇒5=f'0 3++h'0
1⇒3f'0+h'0=5 ;equationiv equationii+ equationiii+ equationiv2f'0+g'0+3g'0+2h'0+3f'0+h'0=6+4+5⇒5f'0+4g'0+3h'0=15 ;equationvequationv-3equationiv⇒5f'0+4g'0+3h'0-9f'0-3h'0=15-15⇒-4f'0+4g'0=0⇒f'0-g'0=0⇒f'0=g'0Putting this value in equationii, we get,2f'0+f'0=6⇒3f'0=6⇒f'0=2=g'0Put this value in equationiii, we get,⇒3×2+2h'0=4⇒2h'0=4-6⇒h'0=-1Putting values of f'0, g'0 and h'0 we get,⇒fgh'0=6×2+3×2+2×-1=12+6-2⇒fgh'0=16 Answer

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let f,g,h are differentiable functions. If f(0 )=1; g(0)=2; h(0)= 3 and the derivatives of their pair wise product at x=0 are- (fg)' (0)=6; (gh)'(0)=4; (hf)'(0)=5 then compute the value of (fgh)'(0)

let f,g,h are differentiable functions. If f(0 )=1; g(0)=2; h(0)= 3 and the derivatives of their pair wise product at x=0 are-

(fg)' (0)=6; (gh)'(0)=4; (hf)'(0)=5

then compute the value of (fgh)'(0)