Question

\(\mathrm{T} / \mathrm{F}:\) The Product Rule states that \(\frac{d}{d x}\left(x^{2} \sin x\right)=2 x \cos x\).

Short answer

False, the correct derivative is \(2x \sin x + x^2 \cos x\).

Step-by-step solution

Understand the Product Rule

The Product Rule is a differentiation rule that applies when you want to find the derivative of the product of two functions. If you have two functions, \(u(x)\) and \(v(x)\), the derivative of their product \(u(x) \cdot v(x)\) is given by \(\frac{d}{dx}\left(u(x) \cdot v(x)\right) = u'(x) v(x) + u(x) v'(x)\).

Identify the Functions in the Product

In the expression \(x^2 \sin x\), let \(u(x) = x^2\) and \(v(x) = \sin x\). These are the two functions we need to differentiate and apply the Product Rule to.

Find Derivatives of Individual Functions

First, find the derivative of \(u(x) = x^2\). The derivative \(u'(x) = 2x\). Then, find the derivative of \(v(x) = \sin x\). The derivative \(v'(x) = \cos x\).

Apply the Product Rule

Using the Product Rule: \(\frac{d}{dx}\left(x^{2} \sin x\right) = u'(x) v(x) + u(x) v'(x)\). This gives us \(2x \cdot \sin x + x^2 \cdot \cos x\).

Compare with the Given Derivative

The derivative according to the Product Rule is \(2x \sin x + x^2 \cos x\). The statement in the question claims that the derivative is \(2x \cos x\), which does not match our result.

Conclusion

The statement is incorrect because the correct derivative is \(2x \sin x + x^2 \cos x\), not just \(2x \cos x\).

Key concepts

Derivatives

Derivatives are a fundamental concept in calculus and are used to describe how a function changes as its input changes. When you take the derivative of a function, you are essentially determining the rate of change or the slope at any given point.
For a function of a single variable, the derivative is typically denoted by \( f'(x) \) or \( \frac{df}{dx} \).

• It measures how much the output of a function changes for a small change in the input.
• Understanding derivatives helps in analyzing and predicting behavior such as motion, growth, and decay.
• Graphically, the derivative at a point can be seen as the slope of the tangent line to the function's graph at that point.

Knowing how to calculate derivatives is crucial in mathematics, physics, and many applied sciences.
It forms the basis for more advanced topics like integration and differential equations.

Differentiation Rules

Differentiation rules are systematic procedures used to find the derivative of functions efficiently. These rules are derived from the limit definition of the derivative but provide shortcuts to avoid lengthy calculations.
Some of the main differentiation rules include:

• Power Rule: For \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
• Sum Rule: The derivative of a sum \( f(x) + g(x) \) is \( f'(x) + g'(x) \).
• Product Rule: Used for products of functions, as elaborated in the solution. It states \( \frac{d}{dx}\left[u(x) \cdot v(x)\right] = u'(x) v(x) + u(x) v'(x) \).
• Quotient Rule: For \( \frac{u(x)}{v(x)} \), the derivative is \( \frac{u'(x) v(x) - u(x) v'(x)}{v(x)^2} \).
• Chain Rule: It applies to compositions, where \( f(g(x)) \) gives \( f'(g(x)) \cdot g'(x) \).

These rules are indispensable tools in calculus for simplifying and solving complex derivatives.

Functions

Functions are mathematical entities that assign every element in one set to exactly one element in another set. They form the backbone of calculus and many mathematical studies.
A function is often expressed as \( f(x) \), where \( x \) is the input and \( f(x) \) is the output.

• Functions can model real-world situations, such as the relationship between time and distance in physics.
• They can be simple, like linear functions \( f(x) = mx + b \), or more complex, like trigonometric or exponential functions.
• The concept involves understanding domain (input values) and range (output values).

This fundamental concept is key for understanding how derivatives and other calculus principles are applied to real-world problems.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides and are essential in mathematics, especially calculus. The basic trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).
Each of these functions has specific properties and identities that are useful in calculus, particularly when dealing with periodic phenomena or rotations.

• Sine Function (\( \sin \theta \)): It gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• Cosine Function (\( \cos \theta \)): It provides the ratio of the adjacent side to the hypotenuse.
• Tangent Function (\( \tan \theta \)): This is the ratio of the opposite side to the adjacent side. It is also the quotient of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

The derivatives of these functions are crucial for solving calculus problems involving oscillatory motion, waves, and other physical phenomena.