First Principles Example 1: x²
Key Questions
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First Principles
#-># Difference Quotient#f'(x)=lim_(h->0)(f(x+h)-f(x))/h# #f(x)=x^2+7x-4# #f(x+h)=(x+h)^2+7(x+h)-4# #f'(x)=lim_(h->0)((x+h)^2+7(x+h)-4-(x^2+7x-4))/h# #f'(x)=lim_(h->0)((x+h)^2+7(x+h)-4-x^2-7x+4)/h# #f'(x)=lim_(h->0)((x+h)^2+7x+7h-4-x^2-7x+4)/h# #f'(x)=lim_(h->0)(x^2+2xh+h^2+7x+7h-4-x^2-7x+4)/h# #f'(x)=lim_(h->0)(2xh+h^2+7h)/h# #f'(x)=lim_(h->0)(h(2x+h+7))/h# #f'(x)=lim_(h->0)(2x+h+7)# #f'(x)=2x+(0)+7# #f'(x)=2x+7# -
Answer:
#f'(x)=2x# Explanation:
#f'(x)=lim_(hto0)(f(x+h)-f(x))/h# #rArrf'(x)=lim_(hto0)((x+h)^2-x^2)/h# #color(white)(rArrf'(x))=lim_(hto0)(cancel(x^2)+2hx+h^2cancel(-x^2))/h# #color(white)(rArrf'(x))=lim_(hto0)(cancel(h)(2x+h))/cancel(h)# #color(white)(rArrf'(x))=2x#
Questions
Derivatives
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Tangent Line to a Curve
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Normal Line to a Tangent
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Slope of a Curve at a Point
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Average Velocity
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Instantaneous Velocity
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Limit Definition of Derivative
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First Principles Example 1: x²
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First Principles Example 2: x³
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First Principles Example 3: square root of x
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Standard Notation and Terminology
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Differentiable vs. Non-differentiable Functions
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Rate of Change of a Function
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Average Rate of Change Over an Interval
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Instantaneous Rate of Change at a Point