 Pamela Meyer

2022-01-13

Find the derivative of a function with new variable
Suppose $f\left(x-3\right)={\left(x-2\right)}^{2}$ and find the ${f}^{\prime }\left({x}^{2}+5\right)$ Jeremy Merritt

Step 1
You calculated the derivative of the composite function
$x↦f\left({x}^{2}+5\right)$,
and evaluated it at x, rather than calculating the derivative of the function $t↦f\left(t\right)$, and evaluating it at ${x}^{2}+5$. Note that by definition
${\left(x↦f\left({x}^{2}+5\right)\right)}^{\prime }\left(x\right)$
$=\underset{h\to 0}{lim}\frac{f\left({\left(c+h\right)}^{2}+5\right)-f\left({x}^{2}+5\right)}{h}$
whereas
${\left(t↦f\left(t\right)\right)}^{\prime }\left({x}^{2}+5\right)=\underset{h\to 0}{lim}\frac{f\left(t+h\right)-f\left(t\right)}{h}{\mid }_{t={x}^{2}+5}$
$=\underset{h\to 0}{lim}\frac{f\left({x}^{2}+5+h\right)-f\left({x}^{2}+5\right)}{h}$ Serita Dewitt

Step 1
As a method to proceed:
${f\left(g\left(x\right)\right)}^{\prime }={g}^{\prime }\left(x\right){f}^{\prime }\left(g\left(x\right)\right)$
Set
$g\left(x\right)={x}^{2}+5$
So,
${g}^{\prime }\left(x\right)=2x$
For x non zero,
${f}^{\prime }\left({x}^{2}+5\right)=\frac{f\left(g\left(x\right)\right)}{2x}$

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