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Question Number 159794 by 0731619 last updated on 21/Nov/21

Commented by cortano last updated on 21/Nov/21

$$\frac{d}{dx}(\int 2^{x}dx)=\frac{d}{dx}\left(x^2\right)$$$$\Rightarrow 2^x=2x$$$$\Rightarrow 2^{x-1}=x$$$$\Rightarrow (x-1)\ln \left(2\right)=\ln \left(x\right)$$$$\Rightarrow x\ln \left(2\right)-\ln \left(x\right)=\ln \left(2\right)$$$$$$

Commented by mr W last updated on 21/Nov/21

$$i{don}^{\prime }tthink{it}^{\prime }scorrectsir.$$$$tosolveanequationf\left(x\right)=g\left(x\right),$$$$itistofindthevalueofx,saya,$$$$whichfulfillstheequation$$$$f\left(a\right)=g\left(a\right).$$$$hereaisaconcretevalue,andyou$$$$cannottreatitasavariable$$$$andderivatew.r.t.toa.$$$$$$$$example:tosolve{x}^3+x^2+1=3x$$$$youcannotdolikethis:$$$$\frac{d}{dx}(x^3+x^2+1)=\frac{d}{dx}\left(3x\right)$$$$3x^2+2x=3$$$$3x^2+2x-3=0$$$$x=\frac{-1\pm \sqrt{10}}{3}$$

Commented by amin96 last updated on 21/Nov/21

$$$$What program do you use to draw graphics?

Commented by mr W last updated on 21/Nov/21

$$whomdoyouask?whichgraphicsdo$$$$youmean?$$

Commented by amin96 last updated on 21/Nov/21

$$$$I mean the funksion graphics. Which app are you watching?

Commented by Tony6400 last updated on 21/Nov/21

Commented by Tony6400 last updated on 21/Nov/21

Commented by Tony6400 last updated on 21/Nov/21

$$Takeu=2^{x}solnu=xln2$$$$\frac{u^{{}^{\prime }}}{u}=ln2and{u}^{{}^{\prime }}=uln2>>>\frac{du}{dx}=2^{x}ln2>>u=ln2\int 2^{x}dx$$$$so\frac{u}{ln2}=\int 2^{x}dx=\frac{2^x}{ln2}$$$$so\frac{2^x}{ln2}=x^2>>>2^x=x^{2}ln2$$$$Nosolution$$

Answered by mr W last updated on 21/Nov/21

$$f\left(x\right)=\int 2^{x}dx=\int e^{x\ln 2}dx=\frac{1}{\ln 2}\int e^{x\ln x}d\left(x\ln 2\right)$$$$=\frac{1}{\ln 2}{e}^{x\ln 2}+C=\frac{2^x}{\ln 2}+C$$$$g\left(x\right)=x^2$$$$sotheequationf\left(x\right)=g\left(x\right)is$$$$\frac{2^x}{\ln 2}+C=x^2$$$$ithasnouniquesolutiondueto$$$$unknownconstantC.$$

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