Problem 24
Question
Find \(f^{\prime}(x)\) $$ f(x)=\sqrt{3 x-\sin ^{2}(4 x)} $$
Step-by-Step Solution
Verified Answer
The derivative is \( f'(x) = \frac{3 - 4 \sin(8x)}{2\sqrt{3x - \sin^2(4x)}} \).
1Step 1: Identify the Function and Apply the Chain Rule
The given function is \( f(x) = \sqrt{3x - \sin^2(4x)} \). This can be rewritten as \( (3x - \sin^2(4x))^{1/2} \). The outer function here is \( u^{1/2} \) where \( u = 3x - \sin^2(4x) \). We will first find the derivative of the outer function using the chain rule.
2Step 2: Differentiate the Outer Function
Differentiate the outer function \( u^{1/2} \) with respect to \( u \) to get \( \frac{1}{2}u^{-1/2} = \frac{1}{2\sqrt{u}} \). Substitute \( u = 3x - \sin^2(4x) \) back into this expression to get \( \frac{1}{2\sqrt{3x - \sin^2(4x)}} \).
3Step 3: Differentiate the Inner Function
Now find the derivative of the inner function \( u = 3x - \sin^2(4x) \). The derivative of \( 3x \) is 3. To differentiate \( -\sin^2(4x) \), use the chain rule: the derivative of \( -\sin^2(4x) \) is \(-2\sin(4x)\cos(4x)\cdot4 \). Simplify this to \(-8\sin(4x)\cos(4x) = -4\sin(8x) \) using the double angle identity.
4Step 4: Combine Derivatives Using Chain Rule
Combine the derivatives from Step 2 and Step 3. According to the chain rule, the derivative \( f'(x) \) is given by the derivative of the outer function times the derivative of the inner function: \[ f'(x) = \frac{1}{2\sqrt{3x - \sin^2(4x)}} \times (3 - 4\sin(8x)). \]
5Step 5: Simplify the Expression
Write the expression for \( f'(x) \) in its simplest form: \[ f'(x) = \frac{3 - 4\sin(8x)}{2\sqrt{3x - \sin^2(4x)}}. \] This is the derivative of the function.
Key Concepts
DerivativeComposite functionTrigonometric differentiation
Derivative
Understanding derivatives is a key part of calculus. Derivatives measure how a function changes as its input changes. For a given function, the derivative tells us the rate of change or the slope of the function at any given point.
To find the derivative of a function, we often rely on a set of rules developed over time. One of the most commonly used is the chain rule. This rule helps us when dealing with composite functions, letting us differentiate them efficiently.
In this problem, we are given a function that is a composition of simpler functions: the square root function and a trigonometric expression inside it. Our aim is to determine how this function changes with respect to changing inputs, requiring the use of the chain rule for accurate differentiation.
To find the derivative of a function, we often rely on a set of rules developed over time. One of the most commonly used is the chain rule. This rule helps us when dealing with composite functions, letting us differentiate them efficiently.
In this problem, we are given a function that is a composition of simpler functions: the square root function and a trigonometric expression inside it. Our aim is to determine how this function changes with respect to changing inputs, requiring the use of the chain rule for accurate differentiation.
Composite function
A composite function is formed by using one function as the input of another. This nesting of functions requires special handling when differentiating. In our example, the function \(f(x) = \sqrt{3x - \sin^2(4x)}\) is a composite because it combines the square root function and a trigonometric polynomial.
To differentiate composite functions, we apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function itself.
To differentiate composite functions, we apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function itself.
- Outer function: \( u^{1/2} \) with respect to \( u \)
- Inner function: \( u = 3x - \sin^2(4x) \)
Trigonometric differentiation
Trigonometric differentiation involves finding the derivatives of trigonometric functions such as sine, cosine, and tangent. These functions are periodic and play a key role in various applications.
In our given function, we encounter the term \(-\sin^2(4x)\). To differentiate this, we initially consider the basic derivative rules like the derivative of \(-\sin(x)\), which is \(-\cos(x)\). However, because our expression is more complex, we use both the chain rule and trigonometric identities.
Specifically, the derivative of \(-\sin^2(4x)\) utilizes the double angle identity: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), allowing further simplification to \(-4\sin(8x)\). This demonstrates the depth required when handling composite trigonometric expressions, blending algebraic manipulation with differentiation techniques.
In our given function, we encounter the term \(-\sin^2(4x)\). To differentiate this, we initially consider the basic derivative rules like the derivative of \(-\sin(x)\), which is \(-\cos(x)\). However, because our expression is more complex, we use both the chain rule and trigonometric identities.
Specifically, the derivative of \(-\sin^2(4x)\) utilizes the double angle identity: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), allowing further simplification to \(-4\sin(8x)\). This demonstrates the depth required when handling composite trigonometric expressions, blending algebraic manipulation with differentiation techniques.
Other exercises in this chapter
Problem 23
Find \(d^{2} y / d x^{2}\) $$ y=\sin x \cos x $$
View solution Problem 23
Find \(d y /\left.d x\right|_{x=1}\) $$ y=(1-x)(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) $$
View solution Problem 24
Use a graphing utility to estimate the value of \(f^{\prime}(1)\) by zooming in on the graph of \(f,\) and then compare your estimate to the exact value obtaine
View solution Problem 24
Find \(d^{2} y / d x^{2}\) $$ y=\tan x $$
View solution