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# 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) = ?"

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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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