Exponential Tangent

A curve increases at a varying rate unless it is an exponential curve this varying rate is called my/mx.An interesting fact to note is that for any curve y=x^n my/mx=e when a=n for example y=x^2 at x=2 my/mx=e-however this doesn't work for y=x^0=1=1^x therefore my/mx always equals 1.An important thing to note is that if dy/dx=0 my/mx=1 a way to think why this works is if a curve is not actually increasing at a point and is staying the same which is what dy/dx=0 represents then at that point it is being multiplied by 1.If dy/dx is negative then my/mx is below 1 but above 0, if dy/dx is positive then my/mx is above 1.The exponential tangent is a curve of exponential growth that has the same my/mx throughout the whole curve as the curve it is the tangent to for a specific point, this means that a curve is being modelled as an exponential function at a point and this also means that dy/dx for both are the same.

The exponential tangent to the curve y=x^n at x=a is y=a(e^((x-a)/a))^n.The proof for this is below:

y=x^n=a^n

y=k(b^x)=k(b^a)

dy/dx=nx^n-1=na^n-1

dy/dx=klnb(b^x)=klnb(b^a)

a^n=k(b^a)

na^n-1=klnb(b^a)

na^n-1/a^n=klnb(b^a)/k(b^a)

na^-1=lnb

e^na^-1=b

y=k(e^na^-1)^x

kb^a=a^n

ke^n=a^n

k=a^n/e^n

y=(a^n/e^n)(e^nx/a)

y=a^n(e^nx/a-n)

y=a^n(e^nx-na/a)

y=a(e^((x-a)/a)^n

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