Question

Find the requested derivative. \(f(x)=x \sin x ;\) find \(f^{\prime \prime}(x)\).

Short answer

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

Step-by-step solution

Find the First Derivative

Start by finding the first derivative of the function \(f(x) = x \sin x\). We will use the product rule, which states that if \(u(x)\) and \(v(x)\) are functions, then the derivative of their product is \((uv)' = u'v + uv'\). Here, let \(u(x) = x\) and \(v(x) = \sin x\), giving \(u'(x) = 1\) and \(v'(x) = \cos x\). Applying the product rule: \[ f'(x) = x \cdot \cos x + 1 \cdot \sin x = x \cos x + \sin x. \]

Find the Second Derivative

Now differentiate the first derivative \(f'(x) = x \cos x + \sin x\) to find \(f''(x)\). Again, we'll use the product rule for the \(x \cos x\) term and a simple derivative for the \(\sin x\) term. Let \(u(x) = x\) and \(v(x) = \cos x\), then \(u'(x) = 1\) and \(v'(x) = -\sin x\). Applying the product rule to \(x \cos x\) gives: \[ (x \cos x)' = 1 \cdot \cos x + x \cdot (-\sin x) = \cos x - x \sin x. \] Adding the derivative of \(\sin x\) (which is \(\cos x\)), we have: \[ f''(x) = \cos x - x \sin x + \cos x = 2 \cos x - x \sin x. \]

Key concepts

Product Rule

The product rule is a fundamental tool you need to know when finding derivatives of products of functions, like in the problem we're exploring. When you have two functions multiplied together, say \( u(x) \) and \( v(x) \), their derivative isn't as simple as taking each derivative separately. This is where the product rule comes in handy. It tells us that the derivative of a product is given by:

• \( (uv)' = u'v + uv' \)

So essentially, you take the derivative of the first function \( u \) and multiply it by the second function \( v \), and then add the product of the first function \( u \) and the derivative of the second function \( v \).

Let's put this into perspective with an example from the exercise where \( f(x) = x \sin(x) \). Here, we let \( u(x) = x \) and \( v(x) = \sin x \).

• Derive \( u(x) \) to get \( u'(x) = 1 \).
• Derive \( v(x) \) to get \( v'(x) = \cos x \).

Now, we apply the product rule:

• \( f'(x) = x \cos x + 1 \cdot \sin x \)

This step gives us the first-hand result of the derivative for our product function. Remember, using the product rule becomes essential when handling functions that multiply together.

Trigonometric Derivatives

Trigonometric derivatives refer to the specific rules for finding derivatives of trigonometric functions. These are crucial when working with functions that include sine, cosine, tangent, etc. For many problems, like our example, you'll need to apply these specific derivatives naturally.When dealing with the trigonometric functions in derivatives:

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

In the problem previously, the function \( f'(x) = x \cos x + \sin x \) includes trigonometric components that must be differentiated again to find the second derivative. This means referencing the basic trigonometric derivative rules.

When we differentiate \( x \cos x + \sin x \) again to obtain \( f''(x) \), we rely on the relationships above, using \( \cos x \)'s derivative (which is \( -\sin x \)) while applying the product rule to the \( x \cos x \) part.

Trigonometric derivative rules act as key tools that help maneuver through differentiating expressions containing \( \sin \) and \( \cos \) effortlessly. By mastering these, differentiating trigonometric functions becomes far more handleable.

Derivative of Sine and Cosine

Differentiating sine and cosine functions is a frequent task in calculus, owing to their constant appearance in mathematical problems. Understanding the derivatives of these basic trigonometric functions can substantially aid in managing more complex differential tasks.The derivatives are standardized, making them easier to remember:

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

These rules allow easier calculation in extended problems. For instance, in our exercise, we first determined \( f'(x) = x \cos x + \sin x \). The derivative \( \sin x \) transforms directly into \( \cos x \). In the portion \( x \cos x \), when taking the second derivative, mindful application of these rules is pivotal.

Considering the application in a broader context, any trigonometric expression containing \( \sin \) or \( \cos \) leverages these derivatives. This reinforces the need to become comfortable with what each derivative represents and how they can be applied effectively in problems.