text: # How do you differentiate  f(x)=(3x^5 - 4x^3 + 2)^23  using the chain rule.?
  
  Dec 9, 2015
  
  $f ' \left(x\right) = 69 {x}^{2} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
  
  #### Explanation:
  
  Remember: Chain rule:
  
  $\text{Derivative of } f \left(g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) \cdot g ' \left(x\right)$
  
  Derivative of Power and chain rule: $f \left(x\right) = {\left(g \left(x\right)\right)}^{n} = f ' \left(x\right) = n \left(g {\left(x\right)}^{n - 1}\right) \cdot g ' \left(x\right)$
  
  Given $f \left(x\right) {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{23}$
  f'(x) = 23(3x^5-4x^3+2)^(23-1) * color(red)(d/(dx)(3x^5 -4x^3+2)
  
  =23(3x^5 -4x^3+2)^22 color(red)((15x^4 -12x^2+0)
  $= 23 {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \textcolor{red}{15 {x}^{4} - 12 {x}^{2}}$ or
  
  by factor out the greatest common factor $\textcolor{b l u e}{3 {x}^{2}}$from $15 {x}^{4} - 12 {x}^{2}$
  $f ' \left(x\right) = 23 \cdot \textcolor{b l u e}{3 {x}^{2}} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
  
  Simplify:
  $f ' \left(x\right) = 69 {x}^{2} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
explanation: The document contains a math question and its corresponding answer. The question asks how to differentiate a specific function using the chain rule, and the answer provides a step-by-step solution to the differentiation problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # How do you differentiate  f(x)=(3x^5 - 4x^3 + 2)^23  using the chain rule.?
  
  Dec 9, 2015
  
  $f ' \left(x\right) = 69 {x}^{2} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
  
  #### Explanation:
  
  Remember: Chain rule:
  
  $\text{Derivative of } f \left(g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) \cdot g ' \left(x\right)$
  
  Derivative of Power and chain rule: $f \left(x\right) = {\left(g \left(x\right)\right)}^{n} = f ' \left(x\right) = n \left(g {\left(x\right)}^{n - 1}\right) \cdot g ' \left(x\right)$
  
  Given $f \left(x\right) {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{23}$
  f'(x) = 23(3x^5-4x^3+2)^(23-1) * color(red)(d/(dx)(3x^5 -4x^3+2)
  
  =23(3x^5 -4x^3+2)^22 color(red)((15x^4 -12x^2+0)
  $= 23 {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \textcolor{red}{15 {x}^{4} - 12 {x}^{2}}$ or
  
  by factor out the greatest common factor $\textcolor{b l u e}{3 {x}^{2}}$from $15 {x}^{4} - 12 {x}^{2}$
  $f ' \left(x\right) = 23 \cdot \textcolor{b l u e}{3 {x}^{2}} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
  
  Simplify:
  $f ' \left(x\right) = 69 {x}^{2} {\left(3 {x}^{5} - 4 {x}^{3} + 2\right)}^{22} \left(5 {x}^{2} - 4\right)$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding answer. The question asks how to differentiate a specific function using the chain rule, and the answer provides a step-by-step solution to the differentiation problem.
  
  Label: question-and-answer