Question

Compute the derivative of the given function. $$f(x)=x^{2} \cos x$$

Short answer

The derivative is \( f'(x) = 2x \cos x - x^2 \sin x \).

Step-by-step solution

Identify the Rule to Use

Since the function is of the form of one function multiplied by another function, use the product rule for derivatives. If you have a function \( u(x) \cdot v(x) \), the derivative \( f'(x) \) is given by the formula: \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \). Here, \( u(x) = x^2 \) and \( v(x) = \cos x \).

Differentiate \( u(x) = x^2 \)

Find the derivative of \( u(x) = x^2 \). Using the power rule for derivatives, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), the derivative is \( u'(x) = 2x \).

Differentiate \( v(x) = \cos x \)

Find the derivative of \( v(x) = \cos x \). The derivative of \( \cos x \) is \( v'(x) = -\sin x \).

Apply the Product Rule

Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \) into the product rule formula.\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]\[ f'(x) = (2x) \cdot (\cos x) + (x^2) \cdot (-\sin x) \].

Simplify the Expression

Write out the expression from the previous step.\[ f'(x) = 2x \cos x - x^2 \sin x \]. This is the derivative of the given function.

Key concepts

Derivative

Understanding derivatives is key when studying calculus. A derivative represents the rate at which a function changes at any particular point. It is essentially the slope of the tangent line to the curve of a function at a given point. For a function like \( f(x) = x^2 \cos x \), finding its derivative involves determining how this function changes as \( x \) varies. The process of finding a derivative can involve different rules, depending on the structure of the function. In this exercise, the product rule is used because the function is a product of two simpler functions: \( x^2 \) and \( \cos x \). Identifying which rule to apply is an important step towards efficiently computing derivatives.

Power Rule

The power rule of differentiation is a fundamental tool for finding derivatives. It applies to functions of the form \( x^n \), where \( n \) is any real number. According to the power rule, the derivative of \( x^n \) is \( n \cdot x^{n-1} \). This means you multiply the original exponent by the coefficient and then decrease the exponent by one.For example, if you differentiate \( x^2 \), you use the power rule:

• The exponent \( n = 2 \)
• Derivative: \( 2 \cdot x^{2-1} = 2x \)

The power rule makes calculating derivatives of polynomial functions straightforward and quick.

Trigonometric Derivatives

Trigonometric functions, such as sine and cosine, have their own special derivatives. These functions come up often in various fields, including physics and engineering.For the cosine function, \( \cos x \), the derivative is \( -\sin x \). The negative sign is crucial because it helps define the wave-like nature of these functions.When differentiating a product of a trigonometric function, like in the function \( f(x) = x^2 \cos x \), trigonometric derivatives blend with other derivative rules, such as the product rule.Remember these basic derivatives for trigonometric functions to make solving calculus problems easier:

• \( \frac{d}{dx} (\sin x) = \cos x \)
• \( \frac{d}{dx} (\cos x) = -\sin x \)

Simplification of Expressions

Simplifying an expression after applying derivative rules is an essential step in calculus. This process refines your result, making it more understandable and often more practical for further applications.Once you have found the derivative using rules like the product rule, you may end up with an expression that can be made simpler or tidier. For the function \( f(x) = x^2 \cos x \), applying the product rule gives:

• \( f'(x) = 2x \cos x - x^2 \sin x \)

Simplification ensures your final answer is clean and free of unnecessary complexity, which can be crucial for problem solving and communicating your results effectively.