lizlozada
Use the chain rule to find the derivative of (2x7)^3.



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myininaya
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So the chain rule is:
dw:1348707512136:dw

myininaya
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f(x) is the outside function
g(x) is the inside function

mathslover
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Should I interrupt?

myininaya
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Which function in this problem is f(x)?

myininaya
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Which function in this problem is g(x)?

lizlozada
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the power of 3 is the outside function, or f(x) and 2x7 is the inside function or g(x)

myininaya
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So f(x)=x^3 and g(x)=2x7
correct :)

mathslover
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how ?

myininaya
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f is the outside function
g is the inside
2x7 is inside the ( )^3

myininaya
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g is inside f

myininaya
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so g is 2x7
f is x^3

myininaya
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\[f(g(x))=f(2x7)=(2x7)^3 \text{ since } f(x)=x^3 \text{ and } g(x)=2x7\]

myininaya
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Now if you want just plug into the formula above that I gave you

lizlozada
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ok so you would set it up x^3(2x7)^2?

myininaya
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The formula known as the chain rule.

myininaya
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If g(x)=2x7, then g'(x)=
If f(x)=x^3, then f'(x)=?

lizlozada
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ohhh 3x^2

myininaya
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And (2x7)'=?

lizlozada
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2

myininaya
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\[(f(g(x))'=f'(g(x))\cdot g'(x)=f'(2x7) \cdot 2\]

myininaya
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Now to find f'(2x7) you need to plug 2x7 into f'

mathslover
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is this the same : f(g(x)) = f(x) g(x)

mathslover
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OK i am interrupting the solution, will take time from @myininaya mam to teach me this,

myininaya
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f(g(x)) means f composed with g

myininaya
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It is not multiplication

lizlozada
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would it be 3x^2(2x7)x2?

myininaya
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No not exactly

lizlozada
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does the 2x7 have an exponent?

myininaya
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f'(x)=3x^2
f'(2x7)= <to find this one replace x with 2x7

lizlozada
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why?

myininaya
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the new input is 2x7

myininaya
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all the outputs have this form 3x^2

myininaya
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except x is the input

myininaya
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and for f'(2x7), the new input is 2x7

lizlozada
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i don't understand...

Shironlom
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dw:1348708652193:dw

Shironlom
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Tip: Try to sense the pattern in order to absorb the chain rule formula if you're looking for them online

lizlozada
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ok so 3(2x7)^2 x 2?

lizlozada
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@Shironlom ?

cgreenwade2000
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Yes.

lizlozada
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thank you!