Pamela Meyer

2022-01-13

Find the derivative of a function with new variable

Suppose$f(x-3)={(x-2)}^{2}$ and find the ${f}^{\prime}({x}^{2}+5)$

Suppose

Jeremy Merritt

Beginner2022-01-14Added 31 answers

Step 1

You calculated the derivative of the composite function

$x\mapsto f({x}^{2}+5)$ ,

and evaluated it at x, rather than calculating the derivative of the function$t\mapsto f\left(t\right)$ , and evaluating it at ${x}^{2}+5$ . Note that by definition

${(x\mapsto f({x}^{2}+5))}^{\prime}\left(x\right)$

$=\underset{h\to 0}{lim}\frac{f({(c+h)}^{2}+5)-f({x}^{2}+5)}{h}$

whereas

$(t\mapsto f\left(t\right))}^{\prime}({x}^{2}+5)=\underset{h\to 0}{lim}\frac{f(t+h)-f\left(t\right)}{h}{\mid}_{t={x}^{2}+5$

$=\underset{h\to 0}{lim}\frac{f({x}^{2}+5+h)-f({x}^{2}+5)}{h}$

You calculated the derivative of the composite function

and evaluated it at x, rather than calculating the derivative of the function

whereas

Serita Dewitt

Beginner2022-01-15Added 41 answers

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,

$f}^{\prime}({x}^{2}+5)=\frac{f\left(g\left(x\right)\right)}{2x$

As a method to proceed:

Set

So,

For x non zero,

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