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Chapter 3: Problem 12

Find the following derivatives. $$\frac{d}{d x}\left(\ln 2 x^{8}\right)$$

Short Answer

Expert verified
Answer: The derivative of the function is \(f'(x) = \frac{8}{x}\).

Step by step solution

01

Identify the outer and inner functions

According to the chain rule, if we have a composite function, say \(g(f(x))\), we need to differentiate the outer function with respect to the inner function and then multiply with the derivative of the inner function. In our case, the outer function is the natural logarithm and the inner function is the power function: Outer Function: \(\ln(u)\), with \(u = 2x^8\). Inner Function: \(2x^8\).
02

Apply the chain rule and differentiate the outer function

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. The derivative of the natural logarithm function is: \(\frac{d}{du}\ln(u) = \frac{1}{u}\). Using the chain rule, we will have the following: $$ \frac{d}{dx}\left(\ln(2x^8)\right) = \frac{1}{u}\frac{d}{dx}(u) = \frac{1}{2x^8}\frac{d}{dx}(2x^8)$$
03

Differentiate the inner function

Now, we need to differentiate the inner function \(2x^8\). To do this, we can use the power rule, which states that if \(h(x)=x^n\), then \(h'(x)=nx^{n-1}\). Applying this rule to the inner function, we get: $$\frac{d}{dx}(2x^8)= 2\cdot 8\cdot x^{8-1} = 16x^7$$
04

Combine the derivatives

Now, we combine the derivatives (Step 2 and Step 3) and simplify the result: $$\frac{d}{d x}\left(\ln 2 x^{8}\right) = \frac{1}{2x^8} \cdot 16x^7$$ $$\frac{d}{d x}\left(\ln 2 x^{8}\right) = \frac{16x^7}{2x^8}$$
05

Simplify the final expression

Finally, we can simplify the final expression and write: $$\frac{d}{d x}\left(\ln 2 x^{8}\right) = \frac{16}{2} \cdot \frac{x^7}{x^8} = 8 \cdot \frac{1}{x} = \frac{8}{x}$$ Therefore, the derivative of the function is: $$\frac{d}{d x}\left(\ln 2 x^{8}\right) = \frac{8}{x}$$

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Most popular questions from this chapter

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

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (lima莽on of Pascal)

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.

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

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=x^{2} e^{3 x}\)
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