Problem 8 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\cos (-5 x) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \cos(-5x) \) is \(-5\sin(5x) \).

Step 1: Identify the Function and the Rule to Use

The given function is \( f(x) = \cos(-5x) \). To find the derivative, we will use the chain rule because this is a composite function. The chain rule states that if you have a function \( f(g(x)) \), its derivative is \( f'(g(x)) \, g'(x) \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( \cos(u) \) with respect to \( u \), where \( u = -5x \). The derivative of \( \cos(u) \) is \(-\sin(u) \). So here, it becomes \(-\sin(-5x) \).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \( u = -5x \) with respect to \( x \). This derivative is \(-5 \).

Step 4: Apply the Chain Rule

According to the chain rule, multiply the derivative of the outer function by the derivative of the inner function. This gives: \(-\sin(-5x) \times (-5) = 5\sin(-5x) \).

Step 5: Simplify the Expression

Since \( \sin(-x) = -\sin(x) \), \( 5\sin(-5x) = -5\sin(5x) \). Thus, the simplified final result is \(-5\sin(5x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. For students trying to grasp this, imagine it as a tool for breaking down complicated expressions into more manageable parts. When you see a function like \( f(x) = \cos(-5x) \), it's actually built using smaller functions.

• Identify the "outer" function, which is \( \cos(u) \).
• Identify the "inner" function, which is \( u = -5x \).

The chain rule tells us to differentiate the outer function first—imagine peeling back layers of an onion—where \( u \) is treated like a simple variable at first. Then, multiply the result by the derivative of the inner function. This approach systematically decodes complex derivatives into simple arithmetic processes, ensuring we don't miss vital steps.

Composite Functions

Composite functions are like nested functions where one function works inside another, similar to Russian dolls.

• In our example, \( \cos(-5x) \) means the equation \( -5x \) is inside the cosine function.
• Understanding the layering of functions helps in applying the chain rule correctly, as you first identify and work with the functions within the outer layers.

When dealing with composite functions, always clearly identify both the inner and outer parts. This clarity helps in applying rules like the chain rule correctly, paving the way for accurate differentiation.

Trigonometric Differentiation

Trigonometric differentiation involves finding derivatives of trigonometric functions like sine, cosine, and tangent. In the exercise, we focus on differentiating \( \cos(-5x) \). Key trigonometric derivatives include:

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

Trigonometric functions have unique properties, like their periodic nature and specific derivative formulas, that you should memorize for quick application. Remember that trigonometric differentiation often goes hand-in-hand with the chain rule, especially in cases involving composite functions. In this example, after differentiating \( \cos(-5x) \), we must remember to apply the derivative of the outer function before handling the multiplication with the inner function's derivative—following through with the final simplification steps to achieve a tidy final result. Be sure to pay attention to these nuances.