# 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
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) = ?"

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 $$\displaystyle{14.2}{e}^{{{0.2}{x}}}.$$

### Relevant Questions

consider the product of 3 functions $$\displaystyle{w}={f}\times{g}\times{h}$$. Find an expression for the derivative of the product in terms of the three given functions and their derivatives. (Remeber that the product of three numbers can be thought of as the product of two of them with the third
$$\displaystyle{w}'=$$?

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
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c. Repeat part (b) for $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$.
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$$f(x)=4x+x^{6}$$
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