作业帮Assume f is a differentiable function whose graph passes thr
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Assume f is a differentiable function whose graph passes through the point (1,4).If g(x) = f(x^2) and the tangent line to the graph of f at (1,4) is y=3x+1,determine g'(x)?
依题目所示,f(x)过点（1,4）且在点（1,4）处的切线为y=3x+1,
因此有f'(x)=3x+1,两边进行积分得到
f(x)=1.5x^2+x+C
由于f(x)过点(1,4),将x=1,y=4代入上式,得C=1.5
故f(x)=1.5x^2+x+1.5
g(x)=f(x^2)=1.5（x^2）^2+x^2+1.5=1.5x^4+x^2+1.5
g'(x)=6x^3+2x
因此有f'(x)=3x+1,两边进行积分得到
f(x)=1.5x^2+x+C
由于f(x)过点(1,4),将x=1,y=4代入上式,得C=1.5
故f(x)=1.5x^2+x+1.5
g(x)=f(x^2)=1.5（x^2）^2+x^2+1.5=1.5x^4+x^2+1.5
g'(x)=6x^3+2x
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