Question

Find \(dy/dx\) for the following functions. $$y=x \cos x \sin x$$

Short answer

Answer: The derivative is: $$\frac{dy}{dx} = -x\sin^2x + x\cos^2x + \cos x\sin x$$

Step-by-step solution

1. Find u'

Since u = x, the derivative with respect to x is: $$u'=1$$ Step 2: Differentiate v

2. Find v'

Since v = cos x, the derivative with respect to x is: $$ v'=-\sin x$$ Step 3: Differentiate w

3. Find w'

Since w = sin x, the derivative with respect to x is: $$w'=\cos x$$ Step 4: Apply the product rule

4. Apply product rule

Using the product rule and the derivatives we have found: $$ \frac{dy}{dx}= (x \cos x \sin x)'=x(-\sin x)(\cos x)+x(\cos x)(\sin x)+1(\cos x)(\sin x)$$ Step 5: Simplify the expression

5. Simplify expression

Combining terms to simplify the expression: $$ \frac{dy}{dx}=-x\sin^2 x + x\cos^2 x + \cos x\sin x$$ The final result is: $$ \frac{dy}{dx} = -x\sin^2 x + x\cos^2 x + \cos x\sin x $$

Key concepts

Understanding the Product Rule in Differentiation

The product rule is an essential concept in calculus, especially when differentiating functions that are the product of two or more factors. In simple terms, when you have a function y that is a product of two functions, say \( u(x) \) and \( v(x) \), the derivative of y with respect to x is given by the product rule: \( (uv)' = u'v + uv' \). This rule extends similarly if there are more than two factors involved.

• First, differentiate each factor independently.
• Next, sum the products of each differentiated term with the others.

Applying this rule requires careful attention to each part of the function, ensuring that derivatives are correctly computed and then multiplied accordingly. Always remember: take one function, differentiate it while keeping others constant, and switch. Then, sum all results to find the complete derivative.

Trigonometric Functions and Their Derivatives

Trigonometric functions, like sine and cosine, play a critical role in calculus. Their derivatives are fundamental for solving many types of problems. Here’s a quick reminder of the basics:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).

Understanding these derivatives is crucial because they recur often in problems involving periodic phenomena or in cases where angles are manipulated.
In problems where trigonometric functions are combined with other functions (like \( y = x \cdot \cos x \cdot \sin x \) from the exercise), remembering these derivatives allows us to apply differentiation rules like the product rule effectively.

The Basics of Derivatives

Derivatives measure how a function changes as its input changes. They are a fundamental concept in calculus, representing the slope of the tangent line to the curve of a function at any point. When finding derivatives, especially of complex expressions, it’s essential to:

• Identify which rules of differentiation apply (product rule, chain rule, etc.).
• Carefully compute limits where necessary.
• Simplify expressions to make them easier to interpret and analyze.

In the given exercise, the function \( y = x \cos x \sin x \) involves a product of three terms, and using the product rule and trigonometric identities can simplify the process of finding its derivative. Thus, derivatives allow you to understand the rate of change in systems modeled by mathematical equations. This understanding is key in both theoretical and applied sciences.