Let $\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x$. Using the definition \(\mathrm{g}'(x)=\lim_{h\to0}\dfrac

Question

Let $\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x$. Using the definition $\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}$, show from first principles that the derivative of $\mathrm{g}(x)=x^2+2x$ is $2x+2$. Prove by induction that, for every positive integer $n$, the $n^{\mathrm{th}}$ derivative of $\mathrm{f}$ is $\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]$.

Accepted Answer

(a) $\mathrm{g}'(x)=2x+2$ (b) $\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]$, $n\ge1$

Quadratic exponential derivative pattern

Quadratic exponential derivative pattern

Solution:

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