Alan

"If  \(log_a b=c\)  then a=?"

\(log_ab=c\Rightarrow a^c=b\Rightarrow a=b^{1/c}\)

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As follows:

\(y=x^3\quad \frac{dy}{dx}=3x^2\\x=y^{1/3}\quad \frac{dx}{dy}=\frac{1}{3}y^{-2/3}\rightarrow \frac{1}{3y^{2/3}} \rightarrow \frac{1}{3(y^{1/3})^2}\rightarrow \frac{1}{3x^2}\)

Now you should be able to see that dy/dx*dx/dy = 1

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Like so:

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Assuming 1GB = 1000MB then possible storage = 8000/3.5 ≈ 2285.7 songs.  However, storage for complete songs is just 2285, which requires 2285*3.5 = 7997.5 MB

So domain (number of songs):  0 to 2285

Range (memory): 0 to 7997.5 MB

(NB memory = 3.5*nbr of songs)

I assume that 'x E i' means x belongs to the integers.  For the smallest value of m we must have the smallest equal increments in x. In other words x must increase by 1 each time.  Thus the first three values of x are -2, -1 and 0, with the corresponding y values being 2, 6 and 10.  Plot these and then extrapolate with a straight line, then plot points where the line crosses the next two values of x to see what k and m must be:

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dy/dx means differentiate y with respect to x

dy/du means differentiate y with respect to u

du/dx means differentiate u with respect to x

y = u²        dy/du = 2u

u = ax + 1  du/dx = a

Since dy/du = 2u and u = ax + 1 we must have dy/du = 2(ax + 1) by simply replacing u on the right hand side by ax + 1.

So dy/du*du/dx = 2(ax + 1)*a → 2a(ax + 1) or 2a²x + 2a

Notice that dy/dx = dy/du*du/dx (this is known as the chain rule) so dy/dx = 2a²x + 2a.   

(See http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-chain-2009-1.pdf for more information on the chain rule.)

Hope this helps!

Like so:

\(y=Ae^{\omega x}+Be^{-\omega x}\\ y'=\omega Ae^{\omega x}-\omega Be^{-\omega x}\\ y''= \omega^2y\)

Now use your initial conditions to find A and B.

Like so:

\(y = \frac{3x-7}{x+1} \text{ Solve for x }\\ y(x+1)=3x-7\\x(y-3)=-(y+7)\\x = -\frac{y+7}{y-3} \text{ swap y and x and replace y by f^{-1}(x) }\\f^{-1}(x)=-\frac{x+7}{x-3}\)

This question was asked and answered here: https://web2.0calc.com/questions/i-need-some-help_8#r1 

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I suspect this is a deliberately awkward question!  None of the integrals listed have results in terms of standard mathematical functions ( except possibly for the ln(x) one; but even there the result is in terms of the imaginary error function, hardly a common function!).  There are solutions in terms of series - see WolframAlpha.