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Chapter 3: Problem 43

Use the General Power Rule where appropriate to find the derivative of the following functions. $$f(x)=\frac{2^{x}}{2^{x}+1}$$

Short Answer

Expert verified
Answer: The derivative of the given function is $$f'(x)=\frac{2^x\ln2(2^x)}{(2^x+1)^2}$$.

Step by step solution

01

Identify the Numerator and Denominator

In the given function $$f(x)=\frac{2^{x}}{2^{x}+1}$$, we have the numerator $$u=2^{x}$$ and the denominator $$v=2^{x}+1$$ Step 2: Differentiate the Numerator
02

Differentiate the Numerator

To find the derivative of the numerator $$u=2^{x}$$, we will use the General Power Rule: $$(a^x)' = a^x \ln a$$, where $$a=2$$ in our case. So, $$(2^x)' = 2^x \ln 2$$ Step 3: Differentiate the Denominator
03

Differentiate the Denominator

To find the derivative of the denominator $$v=2^{x}+1$$, we will differentiate each term separately. The second term is $$1$$, which is a constant, so its derivative is $$0$$. For the first term, we already found the derivative in step 2, which is $$2^x\ln2$$. Combining, we have: $$(2^x+1)'=2^x\ln2+0=2^x\ln2$$ Step 4: Apply the Quotient Rule
04

Apply the Quotient Rule

Now, we will apply the Quotient Rule to find the derivative of the given function:$$\left(\frac{u}{v}\right)' = \frac{u'v-uv'}{v^2}$$Let's substitute the expressions we found in step 2 and step 3 for $$u'$$ and $$v'$$ respectively, and the expressions from step 1 for $$u$$ and $$v$$:$$f'(x) = \frac{(2^x\ln2)(2^x+1)-(2^x)(2^x\ln2)}{(2^x+1)^2}$$ Step 5: Simplify the Expression
05

Simplify the Expression

Next, we will simplify the expression:$$f'(x) = \frac{2^x\ln2(2^x+1-1)}{(2^x+1)^2}=\frac{2^x\ln2(2^x)}{(2^x+1)^2}$$ So, the derivative of the given function is $$f'(x)=\frac{2^x\ln2(2^x)}{(2^x+1)^2}$$.

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Most popular questions from this chapter

Special Quotient Rule In general, the derivative of a quotient is not the quotient of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f / g\) equals \(f^{\prime} / g^{\prime}\).

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of \(a\) and an outer radius of \(b\) is \(V=\pi^{2}(b+a)(b-a)^{2} / 4\) a. Find \(d b / d a\) for a torus with a volume of \(64 \pi^{2}\). b. Evaluate this derivative when \(a=6\) and \(b=10\)

Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.

An observer is \(20 \mathrm{m}\) above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is \(20 \mathrm{m}\) horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of \(5 \mathrm{m} / \mathrm{s}\), what is the rate of change of the angle of elevation when the elevator is \(10 \mathrm{m}\) above the ground? When the elevator is \(40 \mathrm{m}\) above the ground?

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{x}{x+2}\)
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