To find f(g(x)), we just substitute x = g(x) in the function f(x) . For example, when f(x) = x 2 and g(x) = 3x - 5, then f(g(x)) = f(3x - 5) = (3x - 5) 2 .

Problem 19 Differentiate. $$ g(x)=\sec ... [FREE SOLUTION] (2024)

Get started for free

Get started for free Log out

Chapter 2: Problem 19

Differentiate. $$ g(x)=\sec x \csc x $$

Short Answer

Expert verified

The derivative of $g(x) = \sec x \csc x$ is 0.

Step by step solution

Recall the product rule

The product rule for differentiation states that if you have a function in the form of a product of two functions, like $u(x)v(x)$, then its derivative is given by $(uv)' = u'v + uv'$. Here, $u = \sec x$ and $v = \csc x$.

Differentiate $\sec x$

Recall that the derivative of $\sec x$ is $\sec x \tan x$, so $u' = \sec x \tan x$.

Differentiate $\csc x$

Recall that the derivative of $\csc x$ is $-\csc x \cot x$, so $v' = -\csc x \cot x$.

Apply the product rule

Using the product rule, we get $(uv)' = u'v + uv'$. Substituting $u = \sec x$, $v = \csc x$, $u' = \sec x \tan x$, and $v' = -\csc x \cot x$, we have \[\frac{d}{dx} (\sec x \csc x) = (\sec x \tan x)(\csc x) + (\sec x)(-\csc x \cot x).\]

Simplify the expression

Simplify the resulting expression to get \[\sec x \csc x \tan x - \sec x \csc x \cot x.\] Since $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$, we can further simplify to \[\sec x \csc x (\frac{\sin x}{\cos x}) - \sec x \csc x (\frac{\cos x}{\sin x}).\] This reduces to \[\frac{\csc x}{\cos x} - \frac{\sec x}{\sin x}.\]

Final simplification

Since $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$, the expression can be simplified to \[\frac{1}{\sin x \cos x} - \frac{1}{\cos x \sin x} = 0.\] Hence, the derivative of $\sec x \csc x$ is 0.

Key Concepts

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

Product Rule

In calculus, the product rule is a fundamental technique for differentiating products of functions. Given two functions, say $ u(x) $ and $ v(x) $, the product rule helps us find the derivative of their product. The rule states:
\[ (uv)' = u'v + uv'\].
This means we find the derivatives of each function separately and then sum their combinations:

First, differentiate $ u(x) $ to get $ u'(x) $.
Second, differentiate $ v(x) $ to get $ v'(x) $.
Multiply $ u'(x) $ by $ v(x) $, and $ u(x) $ by $ v'(x) $ respectively.
Lastly, sum these two products to get the final derivative of $ u(x)v(x) $.

In our given problem, we have:
\[ u(x) = \sec x \] and \[ v(x) = \csc x \].
By applying the product rule correctly, we can successfully differentiate functions that are products.

Trig Functions Derivatives

Trig functions like $ \sec x $ and $ \csc x $ often appear in calculus problems. Knowing their derivatives is key to solving such exercises. The derivatives are:

The derivative of $ \sec x $ is $ \sec x \tan x $.
The derivative of $ \csc x $ is $ -\csc x \cot x $.

To break this down:

For $ \sec x $, we use its trigonometric identity and rules to get $ \sec x \tan x $.
Similarly, for $ \csc x $, we apply trigonometric differentiation rules yielding $ -\csc x \cot x $.

Using these derivative rules, we help transition smoothly into applying the product rule. For instance, in our problem, we obtain:
\[ u' = \sec x \tan x \] and \[ v' = -\csc x \cot x \].
Knowing these rules may seem tricky, but practice makes it easier!

Simplification in Calculus

Simplification is critical in calculus to get neat and manageable results. After applying the product rule and finding the derivatives of trigonometric functions, simplification helps in presenting the final answer clearly.
For the given problem, let's see step-by-step how simplification works:

First, use the product rule on $ \sec x \csc x $, leading to
\[ (\sec x \tan x)(\csc x) + \sec x(-\csc x \cot x). \]
Second, distribute and simplify, giving
\[ \sec x \csc x \tan x - \sec x \csc x \cot x. \]
Replacing with trig identities, since $ \tan x = \frac{\sin x}{\cos x} $ and $ \cot x = \frac{\cos x}{\sin x} $, we get
\[ \sec x \csc x \frac{\sin x}{\cos x} - \sec x \csc x \frac{\cos x}{\sin x}. \]
Simplifying further, we get
\[ \frac{\csc x}{\cos x} - \frac{\sec x}{\sin x}. \]

Ultimately, knowing $ \sec x = \frac{1}{\cos x} $ and $ \csc x = \frac{1}{\sin x} $, we simplify the expression to
\[ \frac{1}{\sin x \cos x} - \frac{1}{\cos x \sin x} = 0. \]
And, hence, the final answer: The derivative of $ \sec x \csc x $ is 0.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

$Problem 19 Differentiate. $$ g(x)=\sec ... [FREE SOLUTION] (3)$

Most popular questions from this chapter

Growth. The population $P$, in thousands, of a small city is given by$$P(t)=10+\frac{50 t}{2 t^{2}+9}$$ where $t$ is the time, in years. a) Find the growth rate. b) Find the population after $8 \mathrm{yr}$. c) Find the growth rate at $t=12 \mathrm{yI}$.Differentiate. $$ f(x)=(5 x-2) \sqrt{3 x+4} $$Differentiate. $$ y=\sqrt{\sec ^{4} x+x} $$Differentiate. $$ g(t)=t(\csc t)(1+\cos t) $$

See all solutions

Recommended explanations on Biology Textbooks

Biological Organisms

Read Explanation

Communicable Diseases

Read Explanation

Microbiology

Read Explanation

Substance Exchange

Read Explanation

Responding to Change

Read Explanation

Organ Systems

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT

$Problem 19 Differentiate. $$ g(x)=\sec ... [FREE SOLUTION] (2024)$

FAQs

How to calculate derivative of f(g(x))? ›

Answer: The chain rule explains that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, the chain rule helps in differentiating *composite functions*.

Know More ›

What is the chain rule for differentiable functions? ›

Chain rule (simple case): Suppose that f(x,y) is a differentiable function of (x,y), and that r(t) is a differentiable parametrized curve in the x-y plane. Then f(r(t)) is a differentiable function of t, and df(r(t))dt=fx(r(t))x′(t)+fy(r(t))y′(t).

Get More Info Here ›

What is the chain rule simplified? ›

The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

What is chain rule basic equation? ›

The chain rule formula is d/dx ( f(g(x) ) = f' (g(x))·g' (x), whereas the product rule formula is d/dx[f(x). g(x)] = f(x) g'(x)+ f'(x) g(x).

Read On ›

What is the logic behind the chain rule? ›

The chain rule calculates this derivative by following the chain of events that occur when we change the input to g and observe the resulting change in the output of f. A change in the input to g (the sphere) first causes a change in the output of g (the cube).

Problem 19 Differentiate. $$ g(x)=\sec ... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

Recall the product rule

Differentiate \(\sec x\)

Differentiate \(\csc x\)

Apply the product rule

Simplify the expression

Final simplification

Key Concepts

Product Rule

Trig Functions Derivatives

Simplification in Calculus

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Biology Textbooks

Biological Organisms

Communicable Diseases

Microbiology

Substance Exchange

Responding to Change

Organ Systems

Study anywhere. Anytime. Across all devices.

Privacy Overview

FAQs

How to calculate derivative of f(g(x))? ›

What is chain rule basic equation? ›