text: ## Wednesday, June 4, 2014
  
  ### BQ #7: Unit V: Derivatives and the Area Problem
  
  Explain in detail where the formula for the difference quotient comes from now that you know. Include all appropriate terminology (secant line, tangent line, h/delta x, etc.).
  
  The difference quotient helps us to find the slope of a tangent line at any point. But a tangent line only touches the graph once. While the secant line touches the graph twice. In order to find the slope of a secant line, we use the slope formula m = (y2-y1)/( x2-x1). So based on the first point on the graph below it is (x,f(x)) and the second point is (x+h, f(x+h)). So now we know what y2,y1,x2,x1 are. So we just plug this into the slope formula, which is m= (f(x+h))-(f(x))/((x+h)-(x)). So on the top nothing simplifies  but the bottom the 1s cancels out, which leaves us h for the denominator.
  
  That will give us the difference quotient: Sometimes the delta x is refer to h.
  
  http://www.infobarrel.com/Find_the_Derivative_of_a_function_by_using_the_difference_quotient_definition
  
  Reference:
explanation: The text extract is a blog post or educational content explaining the concept of the difference quotient in calculus. It describes the origin of the difference quotient formula and how it is used to find the slope of a tangent line. The extract does not contain a specific math question or an answer to a question; it is more of an explanatory passage.
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distilabel_metadata.role: system
distilabel_metadata.content: ## Wednesday, June 4, 2014
  
  ### BQ #7: Unit V: Derivatives and the Area Problem
  
  Explain in detail where the formula for the difference quotient comes from now that you know. Include all appropriate terminology (secant line, tangent line, h/delta x, etc.).
  
  The difference quotient helps us to find the slope of a tangent line at any point. But a tangent line only touches the graph once. While the secant line touches the graph twice. In order to find the slope of a secant line, we use the slope formula m = (y2-y1)/( x2-x1). So based on the first point on the graph below it is (x,f(x)) and the second point is (x+h, f(x+h)). So now we know what y2,y1,x2,x1 are. So we just plug this into the slope formula, which is m= (f(x+h))-(f(x))/((x+h)-(x)). So on the top nothing simplifies  but the bottom the 1s cancels out, which leaves us h for the denominator.
  
  That will give us the difference quotient: Sometimes the delta x is refer to h.
  
  http://www.infobarrel.com/Find_the_Derivative_of_a_function_by_using_the_difference_quotient_definition
  
  Reference:
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a blog post or educational content explaining the concept of the difference quotient in calculus. It describes the origin of the difference quotient formula and how it is used to find the slope of a tangent line. The extract does not contain a specific math question or an answer to a question; it is more of an explanatory passage.
  
  Label: no-question