First Principles Example 2: x³
Key Questions
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First Principles
#-># Difference Quotient#f'(x)=lim_(h->0)(f(x+h)-f(x))/h# #f(x)=x^3-2x^2+x/4+6# #f(x+h)=(x+h)^3-2(x+h)^2+(x+h)/4+6# #f'(x)=lim_(h->0)((x+h)^3-2(x+h)^2+(x+h)/4+6-(x^3-2x^2+x/4+6))/h# #f'(x)=lim_(h->0)((x+h)^3-2(x+h)^2+(x+h)/4+6-x^3+2x^2-x/4-6)/h# #f'(x)=lim_(h->0)((x+h)^3-2(x+h)^2+(x+h)/4-x^3+2x^2-x/4)/h# #f'(x)=lim_(h->0)((x+h)^3-2(x^2+2xh+h^2)+(x+h)/4-x^3+2x^2-x/4)/h# #f'(x)=lim_(h->0)((x+h)^3-2x^2-4xh-2h^2+(x+h)/4-x^3+2x^2-x/4)/h# #f'(x)=lim_(h->0)((x+h)^3-4xh-2h^2+(x+h)/4-x^3-x/4)/h# #f'(x)=lim_(h->0)((x+h)(x^2+2xh+h^2)-4xh-2h^2+x/4+h/4-x^3-x/4)/h# #f'(x)=lim_(h->0)((x+h)(x^2+2xh+h^2)-4xh-2h^2+h/4-x^3)/h# #f'(x)=lim_(h->0)(x^3+2x^2h+xh^2+hx^2+2xh^2+h^3-4xh-2h^2+h/4-x^3)/h# #f'(x)=lim_(h->0)(2x^2h+xh^2+hx^2+2xh^2+h^3-4xh-2h^2+h/4)/h# #f'(x)=lim_(h->0)(h*(2x^2+xh+x^2+2xh+h^2-4x-2h+1/4))/h# #f'(x)=lim_(h->0)2x^2+xh+x^2+2xh+h^2-4x-2h+1/4# #f'(x)=2x^2+x(0)+x^2+2x(0)+(0)^2-4x-2(0)+1/4# #f'(x)=2x^2+x^2-4x+1/4# #f'(x)=3x^2-4x+1/4# -
First Principles
#-># Difference Quotient#f'(x)=lim_(h->0)(f(x+h)-f(x))/h# #f(x)=x^3# #f(x+h)=(x+h)^3# #f'(x)=lim_(h->0)((x+h)^3-x^3)/h# #f'(x)=lim_(h->0)((x+h)(x^2+2xh+h^2)-x^3)/h# #f'(x)=lim_(h->0)(x^3+2x^2h+xh^2+x^2h+2xh^2+h^3-x^3)/h# #f'(x)=lim_(h->0)(2x^2h+xh^2+x^2h+2xh^2+h^3)/h# #f'(x)=lim_(h->0)(h*(2x^2+xh+x^2+2xh+h^2))/h# #f'(x)=lim_(h->0)2x^2+xh+x^2+2xh+h^2# #f'(x)=lim_(h->0)3x^2+xh+2xh+h^2# #f'(x)=3x^2+x(0)+2x(0)+(0)^2# #f'(x)=3x^2#
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