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differentiate (1 Viewer)

EinstenICEBERG

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y = 5^x Hint: 5= e^ln5

i dont get the hint .... and i dont know how to differentiate it
 

SpiralFlex

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Notice and are each others' inverse function

In general,




The hint is quite misleading (in a sense)





Using your exponential differentiation,



 
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Carrotsticks

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Oh and as a general rule:



For some a > 0
 
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barbernatorr

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the long way around the differential,

y = a^x
ln(y) = ln(a^x)
= x ln(a)
therefore,
x = ln(y) / ln(a)
dx/dy = 1/(y ln(a))
dy/dx = y ln(a)

therefore,
dy/dx = a^x (ln(a))

sorry for no latex, but yeh
 

Carrotsticks

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the long way around the differential,

y = a^x
ln(y) = ln(a^x)
= x ln(a)
therefore,
x = ln(y) / ln(a)
dx/dy = 1/(y ln(a))
dy/dx = y ln(a)

therefore,
dy/dx = a^x (ln(a))

sorry for no latex, but yeh
I actually prefer this way because it is a useful technique for many other differentiating questions, whereas the method we used here does not always work.
 

deswa1

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Also for 4U students who can implicitly differentiate, this also comes in handy for a few types of questions:
<img src="http://latex.codecogs.com/gif.latex?y=a^x\\ lny=xlna\\ (\frac{1}{y})(\frac{dy}{dx})=lna\\ \frac{dy}{dx}=(y)lna\\ \frac{dy}{dx}=(a^x)lna" title="y=a^x\\ lny=xlna\\ (\frac{1}{y})(\frac{dy}{dx})=lna\\ \frac{dy}{dx}=(y)lna\\ \frac{dy}{dx}=(a^x)lna" />
 

barbernatorr

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Also for 4U students who can implicitly differentiate, this also comes in handy for a few types of questions:
<img src="http://latex.codecogs.com/gif.latex?y=a^x\\ lny=xlna\\ (\frac{1}{y})(\frac{dy}{dx})=lna\\ \frac{dy}{dx}=(y)lna\\ \frac{dy}{dx}=(a^x)lna" title="y=a^x\\ lny=xlna\\ (\frac{1}{y})(\frac{dy}{dx})=lna\\ \frac{dy}{dx}=(y)lna\\ \frac{dy}{dx}=(a^x)lna" />
this ^. Implicit differentiation should be taught to 3 unit students as well IMO
 

Timske

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I remember doing implicit diff in a 3u booklet
 

SpiralFlex

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ALSO @TIMSKE YOU GOT PHYSICS TOMORROW. WHY ARE YOU ON?!
 

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