/*! 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 3 Let \(f(x)=\frac{\tan (x)-5}{\se... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 3

Let \(f(x)=\frac{\tan (x)-5}{\sec (x)} .\) Find the following: $$ \begin{array}{l} \text { 1. } f^{\prime}(x)= \\ \text { 2. } f^{\prime}(1)= \end{array} $$

Short Answer

Expert verified
1. \( f^{\prime}(x) = \cos(x) + 5\sin(x) \) \ 2. \( f^{\prime}(1) = \cos(1) + 5\sin(1) \)

Step by step solution

01

Rewrite the function

Start by rewriting the given function. Recall that \ \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \ \). So, \( f(x) = \frac{\frac{\sin(x)}{\cos(x)} - 5}{\frac{1}{\cos(x)}} \).
02

Simplify the function

Simplify the expression obtained in step 1: \ \( f(x) = \frac{\frac{\sin(x)}{\cos(x)} - 5}{\frac{1}{\cos(x)}} = (\sin(x) - 5\cos(x)) \).\ So, \( f(x) = \sin(x) - 5\cos(x) \).
03

Find the derivative

Now, find the derivative \( f^{\prime}(x) \): \ \( f(x) = \sin(x) - 5\cos(x) \). \ Using derivative rules: \ \( f^{\prime}(x) = \cos(x) - 5(-\sin(x)) = \cos(x) + 5\sin(x) \).
04

Evaluate \( f^{\prime}(1) \)

To find \( f^{\prime}(1) \), substitute \( x = 1 \) into the derivative: \ \( f^{\prime}(1) = \cos(1) + 5\sin(1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

calculus problems
Calculus problems can sometimes look intimidating at first glance, but by breaking them down into smaller steps, they become much more manageable. In this exercise, we are given the function:
  • \(f(x) = \frac{\tan(x) - 5}{\frac{1}{\text{sec}(x)}} \)
. The goal is to find its derivative, \(f^{\text{prime}}(x)\), and then evaluate it at \(x = 1\). The process involves several important calculus concepts, including simplifying the function and using rules for finding derivatives.
trigonometric functions
Trigonometric functions like sine, cosine, tangent, and secant play a crucial role in many calculus problems. In this function:
  • the numerator is \( \tan(x)\) which we know is equal to \( \frac{\text{sin}(x)}{\text{cos}(x)} \)
  • and the denominator involves secant, which is \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \)
. Rewriting these trigonometric expressions in their sine and cosine forms makes it simpler to combine and simplify them. It also prepares the function for easy differentiation later on.
finding derivatives
Finding derivatives involves applying rules like the power rule, product rule, quotient rule, and chain rule to obtain the rate of change of a function. Here鈥檚 how we approached it:
  • Firstly, we simplified the given function \( f(x)\) to \( f(x)= \text{sin}(x) - 5\text{cos}(x) \)
  • Then, we found its derivative using basic derivative rules: \( f^{\text{prime}}(x) = \text{cos}(x) - 5(-\text{sin}(x)) = \text{cos}(x) + 5\text{sin}(x) \)
    • This is achieved by remembering that the derivative of \( \text{sin}(x)\) is \( \text{cos}(x)\), and the derivative of \( \text{cos}(x)\) is \( -\text{sin}(x)\). Combining these results gives us the final derivative function.
function simplification
Simplifying a function can make finding its derivative much easier. Initially, we had \( f(x) = \frac{\tan(x) - 5}{\text{sec}(x)}\), which might appear complex. By first converting tangent and secant to their sine and cosine values, and then simplifying, we get to the much simpler function:
  • \( f(x) = \text{sin}(x) - 5\text{cos}(x) \)
. This simplification turns a challenging differentiation problem into a straightforward application of basic derivative rules.

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Most popular questions from this chapter

Implicit differentiation enables us a different perspective from which to see why the rule \(\frac{d}{d x}\left[a^{x}\right]=a^{x} \ln (a)\) holds, if we assume that \(\frac{d}{d x}[\ln (x)]=\frac{1}{x} .\) This exercise leads you through the key steps to do so. a. Let \(y=a^{x}\). Rewrite this equation using the natural logarithm function to write \(x\) in terms of \(y\) (and the constant \(a\) ). b. Differentiate both sides of the equation you found in (a) with respect to \(x\), keeping in mind that \(y\) is implicitly a function of \(x\). c. Solve the equation you found in (b) for \(\frac{d y}{d x},\) and then use the definition of \(y\) to write \(\frac{d y}{d x}\) solely in terms of \(x\). What have you found?

Find the derivative of the function \(h(w)\), below. It may be to your advantage to simplify before differentiating. \(h(w)=5 w \arcsin w\). \(h(w)=5 w \arcsin w\) \(h^{\prime}(w)=\) ______

Given \(F(2)=3, F^{\prime}(2)=4, F(4)=1, F^{\prime}(4)=5\) and \(G(1)=3, G^{\prime}(1)=4, G(4)=2, G^{\prime}(4)=7\) find each of the following. (Enter dne for any derivative that cannot be computed from this information alone.) A. \(H(4)\) if \(H(x)=F(G(x))\) _________ B. \(H^{\prime}(4)\) if \(H(x)=F(G(x))\) __________ C. \(H(4)\) if \(H(x)=G(F(x))\) ________ D. \(H^{\prime}(4)\) if \(H(x)=G(F(x))\) ________ E. \(H^{\prime}(4)\) if \(H(x)=F(x) / G(x)\) ___________

Find the derivative of \(h(t)=t \tan t+\sin t\) \(h^{\prime}(t)=\) _________

Let \(u(x)\) be a differentiable function. For each of the following functions, determine the derivative. Each response will involve \(u\) and / or \(u^{\prime}\). a. \(p(x)=e^{u(x)}\) b. \(q(x)=u\left(e^{x}\right)\) c. \(r(x)=\cot (u(x))\) d. \(s(x)=u(\cot (x))\) e. \(a(x)=u\left(x^{4}\right)\) f. \(b(x)=u^{4}(x)\)
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