/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Verify the following derivative ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 30

Verify the following derivative formulas using the Quotient Rule. $$\frac{d}{d x}(\sec x)=\sec x \tan x$$

Short Answer

Expert verified
Question: Verify the formula for the derivative of the secant function: \(\frac{d}{dx}(\sec x) = \sec x \tan x\) Answer: The given formula has been verified, as we have shown that \(\frac{d}{dx}(\sec x) = \sec x \tan x\).

Step by step solution

01

Rewrite secant function as a quotient

The secant function can be defined as the reciprocal of the cosine function. So, \(\sec x = \frac{1}{\cos x}\).
02

Apply the Quotient Rule

We will now apply the Quotient Rule to the expression \(\frac{1}{\cos x}\), treating \(u(x) = 1\) and \(v(x) = \cos x\). We will also need the derivatives of \(u(x)\) and \(v(x)\) with respect to x, which are \(\frac{d}{dx}(1) = 0\) and \(\frac{d}{dx}(\cos x) = -\sin x\). Applying the Quotient Rule: $$\frac{d}{dx}\big(\frac{1}{\cos x}\big) = \frac{\cos x\cdot 0 - 1\cdot(-\sin x)}{(\cos x)^2} = \frac{\sin x}{(\cos x)^2}$$
03

Simplify the expression and verify the formula

To verify the given formula, we need to show that \(\frac{\sin x}{(\cos x)^2} = \sec x \tan x\). Recall that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). So, the product of \(\sec x\) and \(\tan x\) can be written as: $$\sec x \tan x = \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \frac{\sin x}{(\cos x)^2}$$ This is the same expression as we obtained for the derivative of \(\sec x\). Therefore, we have verified the given formula: $$\frac{d}{dx}(\sec x) = \sec x \tan x$$

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Most popular questions from this chapter

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

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=\frac{x}{x+5}$$

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. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this implies that the rule is true for \(k=n+1 .\) (Hint: Write \(e^{(n+1) x}\) as the product of two functions, and use the Product Rule.)

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