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The first derivative of a function measures how the function's output changes as its input changes. In other words, it shows the rate of change or the slope of the function at any point.
For the given function \( y = 3x + 5 \, the first derivative, \ ( dy/dx \), is found using differentiation. Differentiating means calculating how the function varies as \( x \) changes.

To find the first derivative, you use basic differentiation rules:

• Constant Rule: The derivative of a constant is 0.
• Power Rule: The derivative of \( x \) to the power of n is n \( x^{n-1} \[dx/dx = d/dx(3x + 5) \] Applying this gives \ ( dy/dx = 3 \).

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. This process helps us understand how a function changes at any given point.

When you differentiate, you look at small changes in the function as its input changes. This provides valuable insights into the function’s behavior, such as finding slopes, rates of change, and identifying maxima or minima.

In the exercise, we found the first derivative by applying rules of differentiation. The function \( y = 3x + 5 \) is linear, so its differentiation is straightforward. After differentiating, we found the first derivative to be a constant value of 3. This tells us that our original function has a constant rate of increase because the derivative is a constant and not dependent on \( x \).

Calculus is the branch of mathematics that studies continuous change. It has two major branches: differential calculus and integral calculus.

Differential Calculus: Focuses on finding the derivative of a function to understand how the function changes over time.
Integral Calculus: Deals with the accumulation of quantities and the areas under curves.

In our exercise, we concentrated on differential calculus by finding the first and second derivatives of the function \( y = 3x + 5 \).

This helps us comprehend the behavior and properties of functions, which is crucial for various applications in science, engineering, and other fields.

The second derivative, in this case, is zero. This indicates that the rate of change of our function's rate of change is constant. In other situations, it can provide information about the concavity of the function or how the rate of change itself changes.