Limit Definition of Derivative
Key Questions
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Limit Definition of
f'(a) f'(a)=lim_{h to 0}{f(a+h)-f(a)}/h or
f'(a)=lim_{x to a}{f(x)-f(a)}/{x-a} -
f(x)=c is a constant function, so its value stays the same regardless of the x-value. In particular,f(x+h)=c .By the definition of the derivative,
f'(x)=lim_{h to 0}{f(x+h)-f(x)}/h =lim_{h to 0}{c-c}/{h} =lim_{h to 0}0 =0 -
Remember that the limit definition of the derivative goes like this:
f'(x)=lim_{h rightarrow0}{f(x+h)-f(x)}/{h} .
So, for the posted function, we have
f'(x)=lim_{hrightarrow0}{m(x+h)+b-[mx+b]}/{h}
By multiplying out the numerator,
=lim_{hrightarrow0}{mx+mh+b-mx-b}/{h}
By cancelling outmx 's andb 's,
=lim_{hrightarrow0}{mh}/{h}
By cancellng outh 's,
=lim_{hrightarrow0}m=m
Hence,f'(x)=m .The answer above makes sense since the derivative tells us about the slope of the tangent line to the graph of
f , and the slope of the linear function (its graph is a line) ism . -
Yes, there is a difference since the first limit is defined at
x=0 , but the second one is not.
I hope that this was helpful.
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