Question

Prove that the derivative of an even function is odd, and that the derivative of an odd function is even.

Short answer

Question: Prove that the derivative of an even function is odd, and the derivative of an odd function is even. Answer: To prove this, we can follow the steps in the provided solution. For Part 1, we first find the derivative of the even function f'(x). Then, we substitute -x for x in the derivative to get f'(-x). Finally, we show that f'(-x) = -f'(x), which implies that the derivative of the even function is odd. For Part 2, we find the derivative of the odd function f'(x), replace x with -x in the derivative to get f'(-x), and then show that f'(-x) = f'(x) to prove that the derivative of the odd function is even.

Step-by-step solution

Find the derivative of the given even function

Using the given definition of an even function f(x) = f(-x), find the derivative of the function with respect to x, denoted as f'(x).

Substitute -x for x in the derivative

Replace x with -x in the expression for f'(x) to obtain f'(-x).

Show that f'(-x) = -f'(x)

Demonstrate that f'(-x) = -f'(x) using the properties of derivatives and the given definition of an even function. If we can show this, then f'(x) is odd, and we have proved the first part of the exercise. Part 2: Proving that the derivative of an odd function is even.

Find the derivative of the given odd function

Using the given definition of an odd function f(x) = -f(-x), find the derivative of the function with respect to x, denoted as f'(x).

Substitute -x for x in the derivative

Replace x with -x in the expression for f'(x) to obtain f'(-x).

Show that f'(-x) = f'(x)

Demonstrate that f'(-x) = f'(x) using the properties of derivatives and the given definition of an odd function. If we can show this, then f'(x) is even, and we have proved the second part of the exercise. By demonstrating both Part 1 and Part 2, we will have proved that the derivative of an even function is odd, and that the derivative of an odd function is even.

Key concepts

Even and Odd Functions

Understanding the nature of even and odd functions is a fundamental aspect of higher mathematics, particularly when analyzing their behavior when differentiated. An even function is characterized by having symmetry about the y-axis, meaning for every point on the function, there is an identical point mirrored across the y-axis. Mathematically, this symmetry is defined by the equation \( f(x) = f(-x) \). Visualize a traditional 'M' or a 'W', which lays perfectly over its own reflection when folded along the y-axis.

An odd function, on the other hand, has a different kind of symmetry — rotational symmetry about the origin. This means that if the function were to be rotated 180 degrees about the origin, it would appear unchanged. The defining property for an odd function is \( f(x) = -f(-x) \), implying that flipping the graph around the origin gives you the same graph but with opposite signs. Think of the shape of the letter 'S', which, when flipped and rotated, fits into its initial shape, except that part above the x-axis is now below, and vice versa.

These symmetries play a critical role when we differentiate such functions, revealing interesting properties about their derivatives.

Derivative of an Even Function

When we dive into calculus and take the derivative of an even function, an intriguing property emerges. To illustrate, let's consider an even function \( f(x) \). By definition, we have \( f(x) = f(-x) \). If we differentiate both sides with respect to \( x \) and then substitute \( -x \) in place of \( x \) in the derivative, we get \( f'(-x) \). An even function's derivative will fulfill the property \( f'(-x) = -f'(x) \) — the hallmark of an odd function.

Why is this significant? It means that the slope of the tangent to the curve of an even function at \( x \) has the same magnitude but opposite sign as the slope of the tangent at \( -x \) (mirroring across the y-axis). This consistency is a powerful tool in understanding the behavior of such a function and can be seen in the way various physical phenomena mirror themselves in a system with even symmetry.

Derivative of an Odd Function

Conversely, when we differentiate an odd function \( f(x) \), we encounter a complementary property. Starting with \( f(x) = -f(-x) \) and differentiating both sides with respect to \( x \) leads us to consider \( f'(-x) \). Here, we find that the derivative of an odd function is itself an even function, as it satisfies the relation \( f'(-x) = f'(x) \).

This result implies that slopes of tangents to the curve on either side of the origin are exactly the same. Just as with even functions, this property of odd functions' derivatives has practical implications, such as predicting the behavior of systems with rotational symmetry. Moreover, understanding this principle helps in solving integrals and differential equations where the symmetry of a function can simplify the process and lead to more straightforward solutions.