- #1
Odette
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Member warned that HW posts must include some effort
Can someone explain it to me step by step?
What's the derivative of sin^3(x+1) ^2?
- Context: High School
- Thread starter Odette
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- Derivative
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SUMMARY
The derivative of the function sin^3((x+1)^2) can be calculated using the chain rule. The function is expressed as f(g(h(x))) where f(g) = g^3, g(h) = sin(h), and h(x) = (x+1)^2. The derivative is computed as df/dx = df/dg|g ⋅ dg/dh|h ⋅ dh/dx, resulting in df/dx = 3(sin((x+1)^2))^2 ⋅ cos((x+1)^2) ⋅ 2(x+1).
PREREQUISITES- Understanding of the chain rule in calculus
- Familiarity with trigonometric functions and their derivatives
- Knowledge of exponentiation and polynomial functions
- Ability to manipulate and simplify algebraic expressions
- Study the application of the chain rule in more complex functions
- Learn about higher-order derivatives and their significance
- Explore trigonometric identities and their derivatives
- Practice derivative calculations involving composite functions
Students studying calculus, mathematics educators, and anyone looking to deepen their understanding of derivatives and the chain rule.
- #2
FactChecker
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Add parenthesis to show clearly what the equation is and what the exponents are applied to.
- #3
- #4
FactChecker
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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)
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)
- #5
Odette
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Thank you!FactChecker said:
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