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 Times a Power Function. $$y=x$$

Short answer

The derivative of \(y = x\) is \(1\).

Step-by-step solution

Identify the function

The function given is linear and represented by the equation: \(y = x\).

Apply the power rule

Since \(x\) can be written as \(x^1\), the power rule for derivatives can be applied, which states that if \(y = x^n\), then the derivative \(y' = nx^{n-1}\).

Compute the derivative

Apply the power rule to differentiate \(x^1\). The derivative is \(1 \cdot x^{1-1} = 1 \cdot x^0 = 1\).

Final answer

The derivative of \(y = x\) is simply a constant \(1\), as the exponent of \(x\) decreased by one and one to any power is still one.

Verification by calculator

Use a scientific calculator or a calculus capable calculator to input the function \(y = x\) and find its derivative, which should confirm the result of \(1\).

Key concepts

Power Rule for Derivatives

The power rule for derivatives is a fundamental technique in calculus, widely used for differentiating functions of the form \(y = x^n\), where \(n\) is a real number. When applying this rule, the derivative of \(y\) with respect to \(x\) is \(ny^{n-1}\). For example, if \(y = x^3\), then the derivative \(y' = 3x^{3-1} = 3x^2\).

In the equation \(y = x\), or more explicitly \(y = x^1\), we can apply the power rule by taking \(n = 1\) and computing \(y' = 1 \cdot x^{1-1} = x^0 = 1\), because any nonzero number raised to the power of zero equals one. This approach simplifies the process of finding derivatives, especially when dealing with polynomial functions, which consist of terms in the power function form.

Linear Functions

Linear functions are the simplest type of polynomial function, and they have the general form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. The graph of a linear function is a straight line. These functions are unique because their rate of change is constant, which is reflected by the slope \(m\).

For the specific case of the function \(y = x\), which can be seen as \(y = 1 \cdot x + 0\), the slope is \(1\) and the y-intercept is \(0\). This means the line crosses the origin (\(0\text{,}0\)) and increases by one unit in the y-direction for each unit increase in the x-direction. When differentiating \(y = x\), the derivative consistently equals the slope, which in this case is the constant \(1\), indicating that the rate of change at any point on the line is uniform.

Calculus Calculator Verification

Calculus calculators or computer algebra systems (CAS) are incredibly useful tools for verifying mathematical computations, especially derivatives. Using these tools, students can confirm their hand-derivations, giving them confidence in their understanding of calculus concepts.

To verify the derivative of the linear function \(y = x\), one can input this function into a calculus calculator. After computation, the calculator will display the derivative, which should match the derived answer of \(1\), reinforcing the correctness of the power rule application. Always remember that while calculators are reliable, understanding the underlying principles and being able to perform derivatives by hand remains essential in learning calculus.