Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=a^{2}$$

Short answer

The derivative of the constant function y = a^2 with respect to x is 0.

Step-by-step solution

Identify the function type

The function given is a constant function, which means the output of the function does not depend on the variable (most commonly denoted as 'x'). In this case, the function is simply defined as a constant, squared, or in other words, a constant function.

Apply the constant rule

The derivative of a constant is always zero. In formal terms, if you have a function such that f(x) = C, where C is any constant, then the derivative of this function with respect to x is 0. Since a squared is a constant, we can apply this rule directly.

Write down the derivative

Therefore, the derivative of the function y = a^2 with respect to x is 0.

Key concepts

Constant Function

Understanding the idea of a constant function is fundamental in calculus. A constant function is one where the output value is the same for any input value of the variable. In mathematical terms, if we let the variable be represented by the letter 'x' and the constant by 'C', the function can be defined as f(x) = C. In this type of function, no matter what value 'x' takes, the output is always 'C'.

For example, if we have a constant function described as f(x) = 5, the output will always be 5 whether x is 1, 100, or -50. In graphical terms, this function represents a horizontal line on a coordinate plane, where the y-coordinate is the constant value and it does not change with different x-coordinates.

Calculus

Calculus is a branch of mathematics that studies how things change. It is divided mainly into two areas: differential calculus and integral calculus. Differential calculus deals with the concept of a derivative, which measures how a function changes as its input changes. Integral calculus, on the other hand, deals with the concept of an integral, which measures the accumulation of quantities. Together, these concepts are used to solve problems involving motion, area, volume, and much more in fields like physics, engineering, economics, and beyond.

Calculus has its own language and symbols, and mastering them is essential for understanding and solving problems in a wide array of scientific and mathematical applications. The key to learning calculus is to understand the basic principles and apply them step by step to solve more complex problems.

Differentiation

Differentiation is a fundamental operation in calculus, referring to the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any point and is a measure of its instantaneous rate of change. Mathematically, it is the limit of the slope of the secant line between two points on the graph of the function as the two points come infinitely close.

In practical terms, if you're driving and looking at your speedometer, the speed your speedometer displays at any moment is akin to finding the derivative of your position with respect to time; it tells you how quickly you're changing your position, which is your instantaneous speed.

Constant Rule

In differentiation, the constant rule is an essential and straightforward principle used when taking the derivative of a constant function. According to this rule, the derivative of any constant value is zero. This makes sense if you consider the geometric interpretation: the graph of a constant function is a horizontal line, and the slope (or steepness) of a horizontal line is zero.

To apply the constant rule, identify if the function is a constant — meaning it does not contain any variables like 'x' or 'y'. If this is the case, no matter what the constant is, the derivative with respect to the variable will be 0. For instance, the derivative of f(x) = a^2, where a is a constant, will be zero because the function's output does not vary with 'x'.