Question Number 103322 by abony1303 last updated on 14/Jul/20

A differentiable function f(x) satisfies  f(x^3 −x^2 +x)=2^(x+1)   for every real number x.  When g(x) is the inverse function of f(x),  find g′(4)?

Commented byabony1303 last updated on 14/Jul/20

pls help

Answered by Worm_Tail last updated on 14/Jul/20

f(x^3 −x^2 +x)=2^(x+1) ⇒  x^3 −x^2 +x=f^(−1) (2^(x+1) )  x^3 −x^2 +x=g(2^(x+1) )  3x^2 −2x+1=2^(x+1) ln(2)g′(2^(x+1) )_(x=1)   3−2+1=4ln(2)g′(4)  2=4ln(2)g′(4)⇒g′(4)=(1/(2ln(2)))

Commented byabony1303 last updated on 14/Jul/20

Thank you sir

Answered by mathmax by abdo last updated on 14/Jul/20

we have f(x^3 −x^2  +x) =2^(x+1)  ⇒f^(−1) (2^(x+1) ) =x^3 −x^2 +x  let 2^(x+1)  =t ⇒e^((x+1)ln2)  =t ⇒(x+1)ln2 =ln(t) ⇒x+1 =((lnt)/(ln2)) ⇒  x =((lnt)/(ln2))−1 ⇒f^(−1) (t) =(((lnt)/(ln2)))^3 −(((lnt)/(ln2)))^2  +((lnt)/(ln2)) =g(t) ⇒  g^′ (t) =(1/((ln2)^3 ))×3(((lnt)^2 )/t) −(1/((ln2)^2 ))×2((lnt)/t) +(1/(tln2)) ⇒  g^′ (4) =(3/(4(ln2)^3 ))(2ln2)^2   −(2/((ln2)^2 ))×((2ln(2))/4) +(1/(4ln2))  =(3/(ln2))−(1/(ln2)) +(1/(4ln2)) =(2/(ln2)) +(1/(4ln2)) =(2+(1/4))×(1/(ln2)) =(9/(4ln2))

Commented by1549442205 last updated on 15/Jul/20

Sir′s the idea is right ,however  mistaked  at third line:f^(−1) (t)=(((lnt)/(ln2))−1)^3 −(((lnt)/(ln2))−1)^2 +((lnt)/(ln2))−1  =(((lnt)/(ln2)))^3 −3(((lnt)/(ln2)))^2 +3((lnt)/(ln2))−1−[(((lnt)/(ln2)))^2 −2((lnt)/(ln2))+1]+((lnt)/(ln2))−1  =(((lnt)/(ln2)))^3 −4(((lnt)/(ln2)))^2 +6((lnt)/(ln2))−2  g′(t)=(1/((ln2)^3 ))×3(((lnt)^2 )/t)−(4/((ln2)^2 ))×2((lnt)/t)+(6/(tln2))  g′(4)=(3/(ln2))−(4/(ln2))+(3/(2ln2))=(1/(2ln2))

Commented bymathmax by abdo last updated on 14/Jul/20

yes  thank you sir...

Commented bymathmax by abdo last updated on 14/Jul/20

i forget −1!