What's the derivative of sin^3(x+1) ^2?

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Discussion Overview

The discussion revolves around finding the derivative of the function sin^3((x+1)^2). Participants seek clarification on the application of the chain rule and the correct interpretation of the expression, including the placement of parentheses.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • Some participants request a step-by-step explanation of the derivative process.
  • There is a suggestion to add parentheses to clarify the structure of the equation and the application of exponents.
  • One participant outlines the application of the chain rule, defining the functions involved and providing the derivative formula.
  • The same participant repeats the chain rule application and provides the derivative expression, indicating the steps taken in the calculation.

Areas of Agreement / Disagreement

Participants generally agree on the need for clarity in the expression and the application of the chain rule, but there is no consensus on the final derivative form as no definitive conclusion is reached.

Contextual Notes

There are limitations regarding the clarity of the original expression due to the absence of parentheses, which may affect the interpretation of the derivative process.

Odette
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Member warned that HW posts must include some effort
Can someone explain it to me step by step?
 
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Add parenthesis to show clearly what the equation is and what the exponents are applied to.
 
Odette said:
Can someone explain it to me step by step?
FactChecker said:
Add parenthesis to show clearly what the equation is and what the exponents are applied to.
 

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You can apply the chain rule repeatedly to (sin((x+1)2))3 = f(g(h(x))), where f(g) = g3, g(h)=sin(h), and h(x)=(x+1)2.

The overall equation is
df/dx = df/dg|g ⋅ dg/dh|h ⋅ dh/dx

df/dg = 3g2; dg/dh = cos(h); dh/dx = 2(x+1).

Substituting in gives df/dx = 3(sin((x+1)2))2 ⋅ cos((x+1)2) ⋅ 2(x+1)
 
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FactChecker said:
You can apply the chain rule repeatedly to (sin((x+1)2))3 = f(g(h(x))), where f(g) = g3, g(h)=sin(h), and h(x)=(x+1)2.

The overall equation is
df/dx = df/dg|g ⋅ dg/dh|h ⋅ dh/dx

df/dg = 3g2; dg/dh = cos(h); dh/dx = 2(x+1).

Substituting in gives df/dx = 3(sin((x+1)2))2 ⋅ cos((x+1)2) ⋅ 2(x+1)
Thank you!
 

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