233lectures-2011-honors-lesspauses - Definition(Definition...

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Definition (Definition of Derivative)Letf:RRbe a real valued functionf(x). Then thederivativef0(x)is given by the following formula if the limit exists:f0(x) = limh0f(x+h)-f(x)h
Definition (Definition of Derivative)Letf:RRbe a real valued functionf(x). Then thederivativef0(x)is given by the following formula if the limit exists:f0(x) = limh0f(x+h)-f(x)hExamplef(x) =x2
Definition (Definition of Derivative)Letf:RRbe a real valued functionf(x). Then thederivativef0(x)is given by the following formula if the limit exists:f0(x) = limh0f(x+h)-f(x)hExamplef(x) =x2limh0f(x+h)-f(x)h= limh0(x+h)2-x2h=x2+ 2hx+h2-x2h
Definition (Definition of Derivative)Letf:RRbe a real valued functionf(x). Then thederivativef0(x)is given by the following formula if the limit exists:f0(x) = limh0f(x+h)-f(x)hExamplef(x) =x2limh0f(x+h)-f(x)h= limh0(x+h)2-x2h=x2+ 2hx+h2-x2h= limh02hx+h2h= 2x+h= 2x
Exponential RuleForf(x) =ex,f0(x) =ex
Exponential RuleForf(x) =ex,f0(x) =exDerivatives of sin(x) and cos(x)sin0(x) = cos(x)cos0(x) =-sin(x)
Exponential RuleForf(x) =ex,f0(x) =exDerivatives of sin(x) and cos(x)sin0(x) = cos(x)cos0(x) =-sin(x)ExampleIff(x) = 5ex-3 sin(x), findf0(x).
Exponential RuleForf(x) =ex,f0(x) =exDerivatives of sin(x) and cos(x)sin0(x) = cos(x)cos0(x) =-sin(x)ExampleIff(x) = 5ex-3 sin(x), findf0(x).f0(x) = (5ex-3 sin(x))0
Exponential RuleForf(x) =ex,f0(x) =exDerivatives of sin(x) and cos(x)sin0(x) = cos(x)cos0(x) =-sin(x)ExampleIff(x) = 5ex-3 sin(x), findf0(x).f0(x) = (5ex-3 sin(x))0= (5ex)0-(3 sin(x))0= 5ex-3 cos(x)
Product and quotient rules(f(x)g(x))0=f0(x)g(x) +f(x)g0(x)f(x)g(x)0=f0(x)g(x)-f(x)g0(x)(g(x))2
Product and quotient rules(f(x)g(x))0=f0(x)g(x) +f(x)g0(x)f(x)g(x)0=f0(x)g(x)-f(x)g0(x)(g(x))2ExampleComputef0(x) forf(x) =ex1+x2.
Product and quotient rules(f(x)g(x))0=f0(x)g(x) +f(x)g0(x)f(x)g(x)0=f0(x)g(x)-f(x)g0(x)(g(x))2ExampleComputef0(x) forf(x) =ex1+x2.
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