Challenge question: Product Rule example with factorisation! If $y=x^m(bx-c)^n$ find $y'$.

 Challenge Question:  If $y=x^m(bx-c)^n$  where $m,n$ are positive integers, and $b,c$ are non-zero real numbers, show that the first derivative equals

$$y'=\dfrac{dy}{dx}=x^{m-1}(c-bx)^{n-1}\bigg( mc -b(m+n)x\bigg)$$

Example of finding constants conditions on $y'$ and $y''$.

Thanks to one of my AP6 students for this solution :-)

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Show that $y=f(x)$ where $f(x)=x^{2n}-cx$ has NO points of inflection, where $n\ge 2 $ is a positive integer and $c\ne 0$ is a real number

 

Properties of curves depending on the sign of $y', y''$.

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Example of inlection point and equation of a tangent.