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Chapter 3: Problem 42

Use the General Power Rule where appropriate to find the derivative of the following functions. $$y=\tan \left(x^{0.74}\right)$$

Short Answer

Expert verified
Answer: The derivative of the function $$y=\tan(x^{0.74})$$ is $$\frac{dy}{dx}=0.74\cdot\sec^2(x^{0.74})\cdot x^{-0.26}.$$

Step by step solution

01

Identify the inner and outer functions

In the given function, $$y=\tan\left(x^{0.74}\right),$$ we can identify two functions: 1. Outer function: $$\tan(u)$$, where $$u$$ is the inner function 2. Inner function: $$u=x^{0.74}$$ Step 2: Find the derivative of the outer function
02

Find the derivative of the outer function

To find the derivative of the outer function $$\tan(u),$$ we can use the fact that $$\frac{d}{du}\tan(u)=\sec^2(u)$$. So, $$\frac{dy}{du}=\sec^2(u)$$ Step 3: Find the derivative of the inner function
03

Find the derivative of the inner function

Now, we need to find the derivative of the inner function $$u=x^{0.74}$$. We can apply the General Power Rule for this, which states that $$\frac{d}{dx}x^n=nx^{n-1}$$, so: $$\frac{du}{dx}=\frac{d}{dx}x^{0.74}=0.74x^{0.74-1}=0.74x^{-0.26}$$ Step 4: Apply the Chain Rule and simplify
04

Apply the Chain Rule and simplify

Now we can apply the Chain Rule to get the derivative of the given function. The Chain Rule states that $$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$, so: $$\frac{dy}{dx}=\sec^2(u)\cdot 0.74x^{-0.26}$$ Since we know that $$u=x^{0.74},$$ we can substitute $$u$$ back into the equation. $$\frac{dy}{dx}=\sec^2(x^{0.74})\cdot 0.74x^{-0.26}$$ So, the derivative of the given function is: $$\frac{dy}{dx}=0.74\cdot\sec^2(x^{0.74})\cdot x^{-0.26}$$

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Most popular questions from this chapter

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

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=\tan x ;(1, \pi / 4)$$

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}-2 x-3, \text { for } x \leq 1 ;(12,-3)$$

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