Product and Quotient Rules
The product and quotient rules are:
ddx(f(x)g(x))=f(x)g′(x)+f′(x)g(x)ddx(f(x)g(x))=g(x)f′(x)−f(x)g′(x)g(x)2
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These are sometimes expressed in terms of u and v, as
ddx(uv)=udvdx+dudxvddx(uv)=vdudx−udvdxv2
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Notice that the vdudx term is positive and the udvdx term
is negative. The way to remember that is that, if u and v are both
positive, then increasing the numerator u will increase the ratio
u/v, while increasing the denominator v will decrease the ration
u/v.
Example 1: To take the derivative of the product x2sin(x),
let u=x2 and v=sin(x). Then u′=2x and v′=cos(x),
hence ddx(x2sin(x))=ddx(uv)=x2cos(x)+2xsin(x).
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Example 2: To take the derivative of tan(x)=sin(x)cos(x),
let u=sin(x) and v=cos(x), so u′=cos(x) and v′=−sin(x).
Then ddx(uv)=vu′−uv′v2=cos(x)cos(x)−(sin(x))(−sin(x))cos2(x),
which simplifies to 1cos2(x)=sec2(x), since sin2(x)+cos2(x)=1. |
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