Problem 21
Question
Find the derivative with respect to the independent variable. $$ f(x)=4 \cos x^{2}-2 \cos ^{2} x $$
Step-by-Step Solution
Verified Answer
The derivative is \( \frac{df}{dx} = -8x\sin(x^2) + 2\sin(2x) \).
1Step 1: Identify the Function Components
The function given is \( f(x) = 4\cos(x^2) - 2\cos^2(x) \). You can see it consists of two parts: \( 4\cos(x^2) \) and \( -2\cos^2(x) \). We will need to differentiate both terms separately.
2Step 2: Differentiate the First Term
The first term is \( 4\cos(x^2) \). Differentiate this using the chain rule where the derivative of \( \cos(u) \) is \( -\sin(u) \), and for \( u = x^2 \), \( \frac{du}{dx} = 2x \). This gives: \( \frac{d}{dx}[4\cos(x^2)] = 4\cdot (-\sin(x^2)) \cdot 2x = -8x\sin(x^2) \).
3Step 3: Differentiate the Second Term
The second term is \( -2\cos^2(x) \). Use the chain rule here as well. The derivative of \( \cos^2(x) \) can be found using the power rule and chain rule: \( \frac{d}{dx}[\cos^2(x)] = 2\cos(x)\cdot(-\sin(x)) = -2\cos(x)\sin(x) \). So, \( \frac{d}{dx}[-2\cos^2(x)] = -2\cdot -2\cos(x)\sin(x) = 4\cos(x)\sin(x) \).
4Step 4: Combine the Derivatives
Combine the derivative results from Steps 2 and 3. The derivative of the function \( f(x) \) is: \[\frac{df}{dx} = -8x\sin(x^2) + 4\cos(x)\sin(x)\].
5Step 5: Simplify the Expression
Notice that \( 4\cos(x)\sin(x) \) can be rewritten using the double angle identity \( \sin(2x) = 2\sin(x)\cos(x) \). Thus, \( 4\cos(x)\sin(x) = 2\sin(2x) \). The simplified derivative becomes: \[\frac{df}{dx} = -8x\sin(x^2) + 2\sin(2x)\].
Key Concepts
Chain RuleTrigonometric DifferentiationDouble Angle Identity
Chain Rule
The chain rule is a fundamental technique in calculus used for differentiating composite functions. Imagine a function inside another function, like our first term in the problem, \(4\cos(x^2)\). Here, \(x^2\) is inside the cosine function. To find the derivative, you first take the derivative of the outer function, keeping the inner function unchanged. For the cosine function, the derivative is \(-\sin(u)\), where \(u\) is \(x^2\) here.
Next, you multiply that by the derivative of the inner function. The derivative of \(x^2\) is \(2x\). So applying the chain rule, the derivative of \(\cos(x^2)\) becomes \(-\sin(x^2)\cdot 2x\). The factor of 4 remains unchanged, multiplying with everything else to give \(-8x\sin(x^2)\).
Next, you multiply that by the derivative of the inner function. The derivative of \(x^2\) is \(2x\). So applying the chain rule, the derivative of \(\cos(x^2)\) becomes \(-\sin(x^2)\cdot 2x\). The factor of 4 remains unchanged, multiplying with everything else to give \(-8x\sin(x^2)\).
- Outer function: \(\cos(u)\)
- Inner function: \(u = x^2\)
- Outside derivative: \(-\sin(u)\)
- Inside derivative: \(2x\)
Trigonometric Differentiation
Trigonometric functions like sine and cosine often require special attention in differentiation due to their periodic and oscillating nature. For the expression \(-2\cos^2(x)\) in our problem, we're using both the power rule and trigonometric differentiation.
To differentiate \(\cos^2(x)\), remember that \(\cos^2(x)\) is really a "power" problem: \((\cos(x))^2\). So begin by applying the power rule to differentiate \((u)^2\), which becomes \(2u\cdot\frac{du}{dx}\). Here, \(u\) is \(\cos(x)\), and \(\frac{du}{dx}\) is the derivative of \(\cos(x)\), which is \(-\sin(x)\).
To differentiate \(\cos^2(x)\), remember that \(\cos^2(x)\) is really a "power" problem: \((\cos(x))^2\). So begin by applying the power rule to differentiate \((u)^2\), which becomes \(2u\cdot\frac{du}{dx}\). Here, \(u\) is \(\cos(x)\), and \(\frac{du}{dx}\) is the derivative of \(\cos(x)\), which is \(-\sin(x)\).
- The derivative of \(\cos^2(x)\) becomes \(2\cos(x)(-\sin(x)) = -2\cos(x)\sin(x)\).
- When considering the whole term \(-2\cos^2(x)\), the derivative becomes \(-2\times -2\cos(x)\sin(x) = 4\cos(x)\sin(x)\).
Double Angle Identity
To simplify trigonometric expressions or derivatives, identities can be very helpful. In this problem, we encounter \(4\cos(x)\sin(x)\), which can be simplified using a well-known identity: the double angle identity for sine.
The double angle identity states that \(\sin(2x) = 2\sin(x)\cos(x)\). This converts the product of sine and cosine into a single trigonometric term with a double angle. By recognizing this identity, we can simplify \(4\cos(x)\sin(x)\) to \(2\sin(2x)\), making the expression easier to work with.
The double angle identity states that \(\sin(2x) = 2\sin(x)\cos(x)\). This converts the product of sine and cosine into a single trigonometric term with a double angle. By recognizing this identity, we can simplify \(4\cos(x)\sin(x)\) to \(2\sin(2x)\), making the expression easier to work with.
- Expression: \(4\cos(x)\sin(x)\)
- Using identity: \(2\sin(2x)\)
Other exercises in this chapter
Problem 21
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Differentiate the functions with respect to the independent variable. \(f(x)=\sin \left(e^{x}\right)\)
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Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=(2 x+1)\left(3 x^{2}-1\right)\), at \(x=1\
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