Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x} \tan F(2 x) $$

Step-by-Step Solution

Answer

\(\sec^2(F(2x)) \cdot F'(2x) \cdot 2\)

Step 1: Identify the function and its components

The expression gives us the derivative of a composite function. We have \(\tan F(2x)\). This involves an outer function (\(\tan u\)) and an inner composite function (\(F(2x)\)).

Step 2: Apply the chain rule

To find the derivative of \(\tan F(2x)\), we apply the chain rule. The chain rule states that if you have a composite function \(u(v(x))\), its derivative is: \((d/dx) u(v(x)) = u'(v(x)) \cdot v'(x)\).

Step 3: Differentiate the outer function

The outer function is \(\tan u\), whose derivative is \((\sec^2 u)\). Applying this derivative to our problem, we have \(\sec^2(F(2x))\).

Step 4: Differentiate the inner function

Now we differentiate the inner function, \(F(2x)\). The derivative of \(F(x)\) with respect to \(x\) is \(F'(x)\). Since we have \(F(2x)\), we need to apply the chain rule again, giving us \(F'(2x) \cdot 2\).

Step 5: Combine the derivatives

Combining both the derivatives from steps 3 and 4, according to the chain rule, the derivative \(D_x an F(2x)\) is \(\sec^2(F(2x)) \cdot F'(2x) \cdot 2\).

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus for finding the derivative of composite functions. It allows us to differentiate a function that is nested within another function.
When we consider a function of the form \(u(v(x))\), the chain rule helps break it down into the derivatives of the outer function \(u\) and the inner function \(v\).
The formula for the chain rule is straightforward:

• Take the derivative of the outer function \(u\), with respect to the inner function \(v(x)\), given as \(u'(v(x))\).
• Then, multiply it by the derivative of the inner function \(v(x)\), which is \(v'(x)\).

In our exercise, the chain rule is applied twice! It helps handle the inner composition of trigonometric and other functions, like in \(\tan F(2x)\), effectively.

Composite Functions

Composite functions are essentially functions within functions. They enable us to build more complex operations by plugging one function into another.
This nesting structure is represented as \(f(g(x))\), where \(g(x)\) is the inner function and \(f\) is the outer function.
A composite function requires special handling when differentiating, often through the use of the chain rule.
Let's take the example given in the problem statement:

• The component functions here are the tangent function (the outer function) and \(F(2x)\) (the inner function).
• The presence of an additional modification within \(F(2x)\), specifically, multiplying \(x\) by 2 makes the inner function slightly more complex.

This layering is why we carefully apply the chain rule to differentiate.

Trigonometric Functions

Trigonometric functions like \(\sin, \cos,\) and \(\tan\) play a vital role in calculus. In differentiation, each of these functions has a specific derivative:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(\tan x\) is \(\sec^2 x\).

In the given exercise, we focus on the tangent function. Differentiating \(\tan\) is a recurring necessity in calculus.
For \(\tan F(2x)\), applying the chain rule gives us \(\sec^2(F(2x))\) as the derivative of the outer function. This reveals the subtleties of dealing with trigonometric layers in composite scenarios.

Derivatives

Derivatives are a core concept in calculus. They measure how a function changes as its input changes. Differentiation systematically finds the rate of change between variables.
Key points to keep in mind regarding derivatives include:

• The derivative of a constant is zero. This reflects the idea that constants do not change.
• Basic differentiation rules (e.g., power rule, product rule) help handle various function types.
• Application of derivatives can involve compound rules like the chain rule when dealing with more intricate or layered functions.

For the problem at hand, differentiating a composite trigonometric function using the chain rule uncovers how these processes interconnect.
All these concepts combine to smoothly solve complex differentiation tasks, like finding \(D_x \, \tan F(2x)\). This results in a product of effective derivative strategies.