Problem 36 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=\sin x \cos x $$

Step-by-Step Solution

Answer

The derivative of \( \sin x \cos x \) is \( \cos 2x \).

Step 1: Identify the Rule

The function you are given is in the form of a product of two functions: \( f(x) = \sin x \cdot \cos x \). When finding the derivative of a product of two functions, you will need to use the product rule.

Step 2: Recall the Product Rule Formula

The product rule states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative of their product is given by: \[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \] In this case, let \( u(x) = \sin x \) and \( v(x) = \cos x \).

Step 3: Differentiate \( u(x) \)

Find the derivative of \( u(x) = \sin x \).The derivative is: \( u'(x) = \cos x \).

Step 4: Differentiate \( v(x) \)

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

Step 5: Apply the Product Rule

Substitute \( u(x) \, \cos x \, \cos x \, v'(x) \, -\sin x \) into the product rule formula:\[ \frac{d}{dx} [\sin x \cos x] = (\cos x) \cdot (\cos x) + (\sin x) \cdot (-\sin x) \] Simplify to obtain:\[ \cos^2 x - \sin^2 x \]

Step 6: Simplify Using Trigonometric Identity

Notice that \( \cos^2 x - \sin^2 x \) can be simplified using the trigonometric identity \( \cos 2x = \cos^2 x - \sin^2 x \).Thus, the derivative simplifies to:\[ \frac{d}{dx} [\sin x \cos x] = \cos 2x \]

Key Concepts

Product Rule

When faced with the task of calculating the derivative of products of two functions, the product rule becomes essential. The product rule is a fundamental principle in calculus that guides you on how to differentiate expressions where two functions are multiplied together. Here's a quick breakdown:

• Let the first function be denoted as \( u(x) \) and the second function as \( v(x) \).
• The product rule formula is: \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).

This means that to find the derivative of a product, you differentiate each function separately, multiply the derivative of the first function by the second function, and add it to the product of the first function by the derivative of the second function. This rule is broadly applicable whenever products of any two differentiable functions need to be differentiated.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the identity are defined. They are often used to simplify complex trigonometric expressions and solving equations. In the context of differentiation, these identities can simplify expressions right before or after differentiation.

• One key identity used is \( \cos^2 x + \sin^2 x = 1 \).
• Another useful identity is the double angle identity: \( \cos 2x = \cos^2 x - \sin^2 x \).

When you're working through derivatives involving sine and cosine functions, like in the problem at hand, recognizing opportunities to use these identities can make your work simpler and more elegant. By transforming expressions into a reduced form, calculations become less cumbersome.

Differentiation

Differentiation is the process of finding the derivative, or the rate at which a function is changing at any given point. This is a core concept in calculus and a crucial skill in solving many mathematical problems. Differentiation has several basic rules, each applicable to different scenarios, and mastering these can greatly enhance problem-solving prowess.

• For single functions like \( f(x) = \sin x \), the derivative is \( f'(x) = \cos x \).
• For \( f(x) = \cos x \), it gives a derivative of \( f'(x) = -\sin x \).

In our exercise, we used these derivatives in combination with the product rule to simplify our differentiation process. Remember that identifying the type of expression you're dealing with and the right rule to apply is essential for efficient and accurate differentiation.