More information about video. To determine the derivative of the exponential function, we need to go back to the limit definition of the derivative.

It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: Euler's formula describes two equivalent ways to move in a circle.

This stunning equation is about spinning around? Yes -- and we can understand it by building on a few analogies: Starting at the number 1, see multiplication as a transformation that changes the number: If they can't think it through, Euler's formula is still a magic spell to them.

While writing, I thought a companion video might help explain the ideas more clearly: It follows the post; watch together, or at your leisure. Euler's formula is the latter: If we examine circular motion using trig, and travel x radians: The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle.

Now let's figure out how the e side of the equation accomplishes it. What is Imaginary Growth? Combining x- and y- coordinates into a complex number is tricky, but manageable. But what does an imaginary exponent mean?

Let's step back a bit. When I see 34, I think of it like this: Regular growth is simple: Imaginary growth is different: It's like a jet engine that was strapped on sideways -- instead of going forward, we start pushing at 90 degrees.e is NOT Just a Number.

Describing e as βa constant approximately β is like calling pi βan irrational number, approximately equal to β. Sure, itβs true, but you completely missed the point. e is NOT Just a Number. Describing e as βa constant approximately β is like calling pi βan irrational number, approximately equal to β.

Sure, itβs true, but you completely missed the point. several web pages intended for students; this seems to be the most popular one.

FONTS FINALLY REPAIRED November Browser adjustments: This web page uses sub scripts, super scripts, and unicode iridis-photo-restoration.com latter may display incorrectly on your computer if you are using an old browser and/or an old operating system.

Free logarithmic equation calculator - solve logarithmic equations step-by-step. 3) Ln (x - 4) + Ln x = Ln 21 Solution: Notice, this time we have a log on both sides. If we write the left side as a single log, we can use the rule that if the logs are equal, the quantity inside must be equal.

Algebra -> Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Rewrite the expression ln (a+b) + 4 ln (a-b) - 3 ln c as a single logarithm ln iridis-photo-restoration.com the function A= Please help?

Thank You! Log On.

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Logarithm and Exponential Questions with Answers and Solutions - Grade 12