derivative of 1 x 2 1

2 min read 19-10-2024

When we talk about derivatives in calculus, we are essentially discussing how a function changes as its input changes. The derivative provides us with the slope of the tangent line to the curve of the function at any given point. In this article, we’ll explore the derivative of the function (f(x) = 1 \cdot x^2 + 1) in detail.

What is the Function?

First, let's clarify the function we are analyzing:

[ f(x) = 1 \cdot x^2 + 1 ]

This simplifies to:

[ f(x) = x^2 + 1 ]

How to Find the Derivative

To find the derivative of a function, we use the rules of differentiation. The most important rule for our function is the Power Rule, which states:

If (f(x) = x^n), then the derivative (f'(x) = n \cdot x^{n-1}).

Applying the Power Rule

Identify the terms:
- The term (x^2) applies the power rule.
- The constant (1) has a derivative of (0) because constants do not change.
Differentiate the function:

[ f'(x) = \frac{d}{dx}(x^2 + 1) = \frac{d}{dx}(x^2) + \frac{d}{dx}(1) ]
- For (x^2), using the Power Rule: [ \frac{d}{dx}(x^2) = 2 \cdot x^{2-1} = 2x ]
- For the constant (1): [ \frac{d}{dx}(1) = 0 ]
Combine the results:

[ f'(x) = 2x + 0 = 2x ]

The Result

Thus, the derivative of the function (f(x) = x^2 + 1) is:

[ f'(x) = 2x ]

Interpretation of the Derivative

The derivative (f'(x) = 2x) tells us about the slope of the function at any point (x):

When (x = 0), (f'(0) = 2 \cdot 0 = 0), which indicates that the function has a horizontal tangent at that point (minimum point).
As (x) increases or decreases, the slope of the tangent line varies linearly. It becomes positive for (x > 0) and negative for (x < 0), indicating that the function is increasing for positive (x) and decreasing for negative (x).

Practical Examples

1. Finding Specific Slopes

Let’s find the slope of the tangent line at specific points:

At (x = 2): [ f'(2) = 2 \cdot 2 = 4 \quad \text{(The slope is 4)} ]
At (x = -3): [ f'(-3) = 2 \cdot (-3) = -6 \quad \text{(The slope is -6)} ]

2. Graphical Representation

Graphically, if you were to plot the function (f(x) = x^2 + 1), it would appear as a parabola that opens upwards with its vertex at the point (0,1). The slope of the tangent at any point on the curve can be derived using (f'(x)).

Conclusion

In conclusion, understanding the derivative of a function is fundamental in calculus. The derivative of (f(x) = x^2 + 1) is (f'(x) = 2x), revealing vital information about the function's behavior. This insight allows us to ascertain how the function behaves at any point, find tangent slopes, and better comprehend the properties of the graph itself.

If you're looking for more complex functions or derivative rules, consider exploring topics such as the product rule, quotient rule, and chain rule, which further enrich the understanding of calculus.