IDNLearn.com: Your go-to resource for finding precise and accurate answers. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.

How do I find the derivative of the following equation:

[tex]y=(\frac{e^x}{x+2} )^7[/tex]

I was lost on this question because after using the chain rule, the exponents still did not match the correct answer for the equation which is:[tex](\frac{e^x}{x+2} )^7(7-\frac{7}{x+2} )[/tex]

If someone could break down the steps, that would be great. I can't seem to find any rule in my notes matching.


Sagot :

Answer:

y' = 7 * (e^(7x) * (x + 1))/((x + 2)^8) = (e^x/(x + 2))^7(7 - 7/(x + 2))

Step-by-step explanation:

The Chain Rule

The chain rule claims that for any composite function, h,  composed by two differentiable functions, f and g, the following is always true:

[tex]h(x) = (f\circ g)(x)\\\\\implies h'(x)=(f'\circ g)(x)\cdot g'(x)[/tex]

The Quotient Rule

The quotient rule claims that for any function, h = (f(x))/(g(x)), the following is always true for g(x) ≠ 0:

[tex]h(x)=\frac{f(x)}{g(x)}\\\\h'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{\left(g(x)\right)^2}[/tex]

Working out the Derivative

We need to find the derivative of the following function:

[tex]y=\left(\frac{e^x}{x+2}\right)^7[/tex]

let f(x) = x^7 and g(x) = (e^x)/(x + 2), then y = f(g(x)), which we can differentiate using the chain rule:

[tex]y' = f'(g(x))\times g'(x)=7\left(g(x)\right)^6\times g'(x)=7\left(\frac{e^x}{x+2}\right)^6\times\left[\frac{e^x}{x+2}\right]'[/tex]

We'll find the derivative of g(x) using the quotient rule:

[tex]\left[\frac{e^x}{x+2}\right]'=\frac{[e^x]'(x+2)-e^x[x+2]'}{\left(x+2\right)^2}=\frac{e^x(x+2)-e^x\times1}{(x + 2)^2}=\frac{e^x\left(x+1\right)}{\left(x+2\right)^2}[/tex]

Substituting back into the equation of y:

[tex]y' = 7\left(\frac{e^x}{x+2}\right)^6\times\frac{e^x(x+1)}{(x + 2)^2}[/tex]

We can simplify this derivative to be in the requested form:

[tex]y'=7\times\frac{e^{6x}}{(x+2)^6}\times\frac{e^x(x+1)}{(x+2)^2}\\\\\to y'=\frac{e^{7x}}{(x+2)^7}\times\frac{7(x + 1)}{x+2}\\\\\to y'=\left(\frac{e^x}{x+2}\right)^7\times\frac{7x+7}{x+2}\\\\\to y'=\left(\frac{e^x}{x+2}\right)^7\times\frac{7x + 7 + 7 - 7}{x + 2}\\\\\to y'=\left(\frac{e^x}{x+2}\right)^7\times\frac{7(x + 2) - 7}{x + 2}\\\\\to y'=\left(\frac{e^x}{x+2}\right)^7\left(7-\frac7{x+2}\right)[/tex]

The derivative of y = (e^x/x + 2)^7 is y' = (e^x/(x + 2))^7(7 - 7/(x + 2))

Answer:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7\left(7 -\dfrac{7}{x+2}\right)[/tex]

Step-by-step explanation:

Given equation:

[tex]y=\left(\dfrac{e^x}{x+2}\right)^7[/tex]

To differentiate the equation, we can use the chain rule.

[tex]\boxed{\begin{array}{c}\underline{\text{Chain Rule for Differentiation}}\\\\\text{If\;\;$y=f(u)$\;\;and\;\;$u=g(x)$\;\;then:}\\\\\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}\end{array}}[/tex]

[tex]\textsf{Let $u$}=\dfrac{e^x}{x+2}[/tex]

Rewrite y in terms of u:

[tex]y=u^7[/tex]

Differentiate y with respect to u:

[tex]\dfrac{\text{d}y}{\text{d}u}=7\cdot u^{7-1} \\\\\\ \dfrac{\text{d}y}{\text{d}u}=7u^6[/tex]

Now, differentiate u with respect to x using the quotient rule.

[tex]\boxed{\begin{array}{c}\underline{\textsf{Quotient Rule for Differentiation}}\\\\\dfrac{\text{d}}{\text{d}x}\left(\dfrac{\text{f}(x)}{\text{g}(x)}\right)=\dfrac{\text{g}(x)\text{f}\:'(x)-\text{f}(x)\text{g}\:'(x)}{\left(\text{g}(x)\right)^2}\end{array}}[/tex]

First, identify f(x) and g(x) and differentiate them separately:

[tex]\text{f}(x)=e^x \implies \text{f}'(x)=e^x[/tex]

[tex]\text{g}(x)=x+2 \implies \text{g}'(x)=1[/tex]

Now, put everything into the quotient rule formula:

[tex]\dfrac{\text{d}u}{\text{d}x}=\dfrac{(x+2) \cdot e^x-e^x\cdot 1}{(x+2)^2} \\\\\\ \dfrac{\text{d}u}{\text{d}x}=\dfrac{xe^x+2e^x-e^x}{(x+2)^2} \\\\\\ \dfrac{\text{d}u}{\text{d}x}=\dfrac{xe^x+e^x}{(x+2)^2} \\\\\\ \dfrac{\text{d}u}{\text{d}x}=\dfrac{e^x(x+1)}{(x+2)^2}[/tex]

Substitute the expressions for dy/du and du/dx into the chain rule formula:

[tex]\dfrac{\text{d}y}{\text{d}x}=7u^6 \cdot \dfrac{e^x(x+1)}{(x+2)^2}[/tex]

Substitute back in u:

[tex]\dfrac{\text{d}y}{\text{d}x}=7\left(\dfrac{e^x}{x+2}\right)^6 \cdot \dfrac{e^x(x+1)}{(x+2)^2}[/tex]

Therefore, the derivative of the given equation is:

[tex]\dfrac{\text{d}y}{\text{d}x}=7\left(\dfrac{e^x}{x+2}\right)^6 \cdot \dfrac{e^x(x+1)}{(x+2)^2}[/tex]

[tex]\dotfill[/tex]

To express the derivative in the preferred form, begin by factoring out [tex]\frac{e^x}{x+2}[/tex] from the second fraction and combine factors:

[tex]\dfrac{\text{d}y}{\text{d}x}=7\left(\dfrac{e^x}{x+2}\right)^6 \cdot \left(\dfrac{e^x}{x+2}\right) \cdot \left(\dfrac{x+1}{x+2} \right) \\\\\\ \dfrac{\text{d}y}{\text{d}x}=7\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(\dfrac{x+1}{x+2} \right)[/tex]

Now, multiply the second fraction by the constant 7:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(\dfrac{7x+7}{x+2} \right) \\\\\\[/tex]

Write the numerator of the second fraction as 7x + 14 - 7:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(\dfrac{7x+14-7}{x+2} \right)[/tex]

Factor out x + 2 from the first two terms of the numerator of the second fraction:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(\dfrac{7(x+2)-7}{x+2} \right) \\\\\\[/tex]

Split into two fractions:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(\dfrac{7(x+2)}{x+2} -\dfrac{7}{x+2}\right) \\\\\\[/tex]

Finally, cancel the common factor (x + 2):

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7 \cdot \left(7 -\dfrac{7}{x+2}\right) \\\\\\[/tex]

Therefore, the derivative of the given equation in the preferred form is:

[tex]\dfrac{\text{d}y}{\text{d}x}=\left(\dfrac{e^x}{x+2}\right)^7\left(7 -\dfrac{7}{x+2}\right)[/tex]

We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.