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Chapter 3: Problem 77

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln (3 x+1)^{4}$$

Short Answer

Expert verified
Answer: The derivative of the function is \(f'(x) = \frac{12}{3x + 1}\).

Step by step solution

01

Rewriting the given function using logarithmic properties

Using the exponent property of logarithms, we can rewrite the given function as follows: $$f(x) = 4\ln(3x + 1)$$ Now we can move on to finding its derivative.
02

Differentiation using chain rule

To find the derivative of \(f(x)\), we'll use the chain rule of differentiation. Recall that the chain rule states that: $$(g\circ h)'(x)=g'(h(x))h'(x).$$ In our case, let \(g(u) = 4\ln(u)\) and \(h(x) = 3x + 1\). Thus, \(f(x) = g\circ h(x)\). Differentiating both \(g(u)\) and \(h(x)\), we get: $$g'(u) = \frac{4}{u}$$ and $$h'(x) = 3$$ Now let's apply the chain rule:
03

Computing the derivative using the chain rule

Applying the chain rule, we get: $$f'(x) = g'(h(x))h'(x) = \frac{4}{h(x)}\times 3$$ Replace \(h(x)\) with the expression we found earlier, \(3x + 1\): $$f'(x) = \frac{4}{3x + 1}\times 3$$ Finally, simplifying the expression, we get:
04

Simplify the derivative expression

$$f'(x) = \frac{12}{3x + 1}$$ So, the derivative of the given function is: $$f'(x) = \frac{12}{3x + 1}$$

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Most popular questions from this chapter

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(x f(x))\right|_{x=3}$$

A hot-air balloon is \(150 \mathrm{ft}\) above the ground when a motorcycle passes directly beneath it (traveling in a straight line on a horizontal road) going \(40 \mathrm{mi} / \mathrm{hr}(58.67 \mathrm{ft} / \mathrm{s})\) If the balloon is rising vertically at a rate of \(10 \mathrm{ft} / \mathrm{s},\) what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of $$f(x)=x^{3} ;(8,2)$$

Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of \(x\) b. Graph the tangent lines on the given graph. \(4 x^{3}=y^{2}(4-x) ; x=2\) (cissoid of Diocles)

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(x y=a ; x^{2}-y^{2}=b,\) where \(a\) and \(b\) are constants
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