Problem 7 - Calculus for Biology and Medicine

Question

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=\cos x ; x=0\)

Step-by-Step Solution

Answer

The derivative of \( y = \cos x \) at \( x = 0 \) is 0.

Step 1: Understand the Problem

To find the derivative of the function at a specific point, we need to understand that the derivative at a point gives the slope of the tangent line to the function at that point. In this case, we need to find the derivative of the function \(f(x) = \cos x\) at \(x = 0\).

Step 2: Differentiate the Function

Differentiate \(f(x) = \cos x\) using the derivative rule for cosine. The derivative of \(\cos x\) is \(-\sin x\). Thus, \(f'(x) = -\sin x\).

Step 3: Evaluate the Derivative at the Point

Substitute \(x = 0\) into the derivative \(f'(x) = -\sin x\) to find \(f'(0)\). Since \(-\sin 0 = 0\), we have \(f'(0) = 0\).

Step 4: Conclusion

The derivative of \(f(x) = \cos x\) at \(x = 0\) is \(f'(0) = 0\), which means that the slope of the tangent line to the curve at this point is zero. This indicates that the tangent is horizontal at this point.

Key Concepts

Differentiation

Differentiation is a core concept in calculus. It involves finding the derivative of a function, which tells us the rate at which the function's output changes with respect to its input. The derivative provides important information, such as the slope of a curve at a given point.
To differentiate a function, we employ specific rules. For example, the derivative of the function \(f(x)\) is often denoted as \(f'(x)\) or \(\frac{df}{dx}\). Standard rules such as the power rule, product rule, and quotient rule are used to differentiate many functions.
In the exercise provided, we differentiate a trigonometric function, \(f(x) = \cos x\). By applying the derivative rule for cosine, we find that the derivative is \(-\sin x\). Understanding how to properly apply these rules is crucial for finding derivatives accurately.

Trigonometric Functions

Trigonometric functions are mathematical functions based on angles and the lengths of a right triangle. These include sine, cosine, and tangent functions, which are essential in analyzing periodic phenomena such as waves.
In calculus, trigonometric functions have well-defined derivatives that are useful in many applications. The derivative of the cosine function is \(-\sin x\), which means that the rate of change of the cosine function is given by the negative sine function.
This relationship is particularly important because it helps model situations in engineering, physics, and various other fields where waveforms and periodic motions are analyzed. Understanding how to differentiate trigonometric functions is a foundational skill in both calculus and applied mathematics.

Tangent Line

A tangent line is a straight line that touches a curve at just one point. It represents the instantaneous rate of change of the function at that point. For any function, the slope of this tangent line at a specific point is given by the derivative evaluated at that point.
In the given problem, the tangent line at \(x = 0\) for the function \(f(x) = \cos x\) was investigated. The derivative \(f'(0) = 0\) indicates that the slope of the tangent is zero at this point.
This means the tangent line is horizontal, lying perfectly parallel to the x-axis at \(x = 0\). Tangent lines provide valuable insights into the behavior of functions and are essential tools for graph analysis and optimization problems.