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Given-f-x-5x-cos-3x-Find-the-value-of-d-dx-f-1-1-

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Question Number 135313 by bobhans last updated on 12/Mar/21
Given f(x) = 5x+cos (3x) Find the value of (d/dx) [f^(−1) (1)]
Givenf(x)=5x+cos(3x)Findthevalueofddx[f−1(1)]
Answered by liberty last updated on 12/Mar/21
(1)f(0)=1⇔f^(−1) (1)=0 (2) (d/dx) [ f^(−1) (1)]=(1/(5−3sin 3x)) ∣_(x = f^(−1) (1)) = (1/(5−3sin 3x))∣_(x = 0) = (1/5)
(1)f(0)=1⇔f−1(1)=0(2)ddx[f−1(1)]=15−3sin3x∣x=f−1(1)=15−3sin3x∣x=0=15
Answered by mathmax by abdo last updated on 12/Mar/21
we know (f^(−1) (x_0 ))^′ =(1/(f^′ (f^(−1) (x_0 )))) ⇒f^(−1) (1))^′ =(1/(f^′ (f^(−1) (1)))) f(0)=1 ⇒f^(−1) (1)=0 ⇒(d/dx)(f^(−1) (1))=(1/(f^′ (0)))=(1/5) f^′ (x)=5−3sinx ⇒f^′ (0)=5
weknow(f−1(x0))′=1f′(f−1(x0))⇒f−1(1))′=1f′(f−1(1))f(0)=1⇒f−1(1)=0⇒ddx(f−1(1))=1f′(0)=15f′(x)=5−3sinx⇒f′(0)=5

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