For the composite function, identify an inside function and an oposite fnction abd write the derivative with respect to x of the composite function. (The function is of the form f(x)=g(h(x)). Use non-identity dunctions for g(h) and h(x).) f(x)=71e^(0.2x) {g(h), h(x)} = ? f'(x) = ?"

For the composite function, identify an inside function and an oposite fnction abd write the derivative with respect to x of the composite function. (The function is of the form f(x)=g(h(x)). Use non-identity dunctions for g(h) and h(x).) f(x)=71e^(0.2x) {g(h), h(x)} = ? f'(x) = ?"

Question
Composite functions
asked 2021-02-06
For the composite function, identify an inside function and an oposite fnction abd write the derivative with respect to x of the composite function. (The function is of the form f(x)=g(h(x)). Use non-identity dunctions for g(h) and h(x).)
\(\displaystyle{f{{\left({x}\right)}}}={71}{e}^{{{0.2}{x}}}\)
{g(h), h(x)} = ?
f'(x) = ?"

Answers (1)

2021-02-07
The given function can be decomposed into two functions, the outer function will be,
\(\displaystyle{g{{\left({h}\right)}}}={71}{e}^{{h}}\)
and the inner function will be,
\(\displaystyle{h}{\left({x}\right)}={0.2}{x}\)
so when we compose these function we get the composite function,
\(\displaystyle{f{{\left({x}\right)}}}={g}{o}{h}{\left({x}\right)}={71}{e}^{{{0.2}{x}}}\)
The formula for differentiation of composite function is,
\(\displaystyle{f}′{\left({x}\right)}={g}′{\left({h}{\left({x}\right)}\right)}·{h}′{\left({x}\right)}\)
substitute \(\displaystyle{g{{\left({h}\right)}}}={71}{e}^{{h}}\) and \(\displaystyle{h}{\left({x}\right)}={0.2}{x}\) into the formula,
\(\displaystyle{f}'{\left({x}\right)}=\frac{{d}}{{{d}{h}{\left({x}\right)}}}{\left({71}{e}^{{{h}{\left({x}\right)}}}\right)}\cdot\frac{{x}}{{\left.{d}{x}\right.}}{\left({0.2}{x}\right)}\)
\(\displaystyle={71}{e}^{{{h}{\left({x}\right)}}}\cdot{0.2}\)
\(\displaystyle={14.2}{e}^{{{0.2}{x}}}\)
hence the differentiation of f(x) is \(\displaystyle{14.2}{e}^{{{0.2}{x}}}.\)
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