91Ó°ÊÓ

The process of finding the derivative of a function is known as differentiation. In essence, this is a way to find the rate at which a quantity changes with respect to another. In mathematical terms, the derivative of a function, represented as \(f'(x)\), gives us the slope of the tangent line to the function at any point \(x\).

In our example, we needed to find the derivative of the function \(f(x) = -4x^2 \cos x\). The given function is the product of two functions, \( u(x) = -4x^2 \) and \( v(x) = \cos x \). To tackle this, we need a technique that works well with products of functions, which is precisely what the product rule provides.

The power rule is a fundamental tool in calculus used to easily find the derivative of a function in the form \( ax^n \). This rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). For a function multiplied by a constant, such as \( ax^n \), the derivative is simply \( f'(x) = nax^{n-1} \).

In this particular exercise, the derivative of \( u(x) = -4x^2 \) is calculated using the power rule. By applying the rule, we take the exponent of \( x \) (which is 2), multiply it with the coefficient \(-4\), resulting in \(-8\), and subtract 1 from the exponent to get \( x^{1} \). Therefore, \( u'(x) = -8x \).

• Power rule helps in easily finding derivatives of polynomial functions.
• The rule simplifies the process of differentiation by providing a quick formula.

Trigonometric functions such as sine and cosine are common in calculus, and their derivatives are important to understand. These derivatives derive from the basic definitions of the functions on the unit circle.

For example, the derivative of \( \cos x \) is \( -\sin x \). This result can be interpreted as the slope of the cosine wave at any given point \( x \). Differentiating trigonometric functions involves understanding their cyclic and oscillatory nature.

• \( \cos x \) derivative is \( -\sin x \), while \( \sin x \) derivative is \( \cos x \).
• The negative sign for \( \cos x \)'s derivative reflects the fact that its slope is negative whenever cos decreases.

In the exercise discussed, we used the derivative of \( \cos x \) in combination with the product rule to differentiate the product \( -4x^2 \cos x \). Recognizing and applying these derivatives is key to solving such problems efficiently.