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Flashcards: Determine Local Linearity and Linearization

Differentiate,

$y = \sin(\sin(\sin\(3x)))$

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \sin(\sin(3x))\times \sin(3x)$

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \cos(\sin(3x))$

$\frac{dy}{dx}=3\cos(\sin(\sin(x)))\times \sin(\sin(x))\times \cos(3x)$

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))$

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Flashcards: Determine Local Linearity and Linearization

Differentiate,

$y = \sin(\sin(\sin\(3x)))$

Your answer:

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \sin(\sin(3x))\times \sin(3x)$

Your answer:

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \cos(\sin(3x))$

Your answer:

$\frac{dy}{dx}=3\cos(\sin(\sin(x)))\times \sin(\sin(x))\times \cos(3x)$

Your answer:

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))$

Correct answer:

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

Explanation:

Differentiate,

$y = \sin(\sin(\sin\(3x)))$

Strategy

This one at first glance appears difficult even if we recognize that the chain rule is needed; we have a function within a function within a function within a function. To avoid making mistakes, it's best to start by defining variables to make the calculation easier to follow.

Let's start with the outermost function, we will write as a function of by setting,

$u = \sin(\sin(3x))$

______________________________________________________

$y = \sin(\underset{u}{ \underbrace{\sin(\sin(3x))}}) = \sin(u)$

$y = \sin(u)$

_______________________________________________________

Similarly, define to write as a function of

$u = \sin(\underset{v}{ \underbrace{\sin(3x)}}) = \sin(v)$

$u = \sin(v)$

_______________________________________________________

Write as a function of

$v = \sin(\underset{w}{ \underbrace{3x}} )=\sin(w)$

$v = \sin(w)$

_______________________________________________________

Finally, define the inner-most function, , as the function of

w=3x

________________________________________________________

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dw}\frac{dw}{dx}$

Since $\frac{dw}{dx}=3$ we will just substitute that in and move to the front.

$\frac{dy}{dx}=3\cos(u)\cos(v)\cos(w)$

That was easy enough, now just write everything in terms of by going back to the definitions of and .

$u = \sin(\sin(3x))$

$v = \sin(3x)$

w = 3x

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

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Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

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All Calculus AB Resources

Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

Flashcards: Determine Local Linearity and Linearization

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DMCA Complaint

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