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Magazine Chapter 3: Problem 21
Find \(d y / d x\) $$ y=\ln (\tan x) $$
Short Answer
Expert verified
The derivative is \( \frac{\sec^2 x}{\tan x} \).
Step by step solution
01
Differentiate using the chain rule
To find the derivative \( \frac{dy}{dx} \) of the function \( y = \ln(\tan x) \), we need to apply the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \ln(u) \) and \( g(x) = \tan x \).
02
Find the derivative of the outer function
The derivative of the natural logarithm function \( f(u) = \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Therefore, the derivative of \( f(g(x)) = \ln(\tan x) \) with respect to \( \tan x \) is \( \frac{1}{\tan x} \).
03
Find the derivative of the inner function
The derivative of \( g(x) = \tan x \) with respect to \( x \) is \( \sec^2 x \). This is a standard derivative result.
04
Apply the chain rule
According to the chain rule, the derivative \( \frac{dy}{dx} \) is the product of the derivative of the outer function and the derivative of the inner function: \[ \frac{dy}{dx} = \frac{1}{\tan x} \cdot \sec^2 x. \]
05
Simplify the expression
Using the identity \( \sec^2 x = 1 + \tan^2 x \), simplify the expression if needed. However, in this case, the expression \( \frac{1}{\tan x} \cdot \sec^2 x \) can also be expressed as \( \frac{\sec^2 x}{\tan x} \), and no further simplification is typically required.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. Composite functions are those that can be expressed as one function inside another, such as \( y = f(g(x)) \). This rule tells us how to differentiate these types of functions.
- To apply the chain rule, we first differentiate the outer function, \( f \), with respect to the inner function \( g(x) \).
- Next, we multiply this by the derivative of the inner function, \( g(x) \), with respect to \( x \).
Derivative of tan x
The derivative of the tangent function \( \tan x \) is a key result needed when applying the chain rule in the given exercise. This derivative is something you'll encounter often, so it's worth committing to memory:
- \( \frac{d}{dx}\tan x = \sec^2 x \)
Natural Logarithm Differentiation
Differentiation of the natural logarithm function \( \ln(u) \) involves a very specific rule: the derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). This simple rule is applied in many calculus problems where a natural logarithm appears.
- For our function, \( f(u) = \ln(u) \), the derivative is \( \frac{1}{u} \).
Secant Squared Function
The secant squared function, denoted as \( \sec^2 x \), plays a crucial role in trigonometric differentiation, especially with tangent functions. It emerges directly from differentiating \( \tan x \).
- Recall that the identity \( \sec x = \frac{1}{\cos x} \) helps define \( \sec^2 x \).
- Essentially, \( \sec^2 x = (\sec x)^2 = \frac{1}{\cos^2 x} \).
