91Ó°ÊÓ

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 \(-8x\sin(x^2)\).

• Outer function: \(\cos(u)\)
• Inner function: \(u = x^2\)
• Outside derivative: \(-\sin(u)\)
• Inside derivative: \(2x\)

Combining these gives the complete derivative for the first part of our problem.

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 \(-\sin(x)\).

• 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 \(-2\times -2\cos(x)\sin(x) = 4\cos(x)\sin(x)\).

Understanding these steps helps handle more complex trigonometric derivatives with ease.

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.

• Expression: \(4\cos(x)\sin(x)\)
• Using identity: \(2\sin(2x)\)

Simplification using trigonometric identities can save time and reduce complexity in calculus problems, allowing for a more straightforward solution.