91Ó°ÊÓ

Derivatives are fundamental in calculus and represent the rate at which a function changes. In simple terms, a derivative gives us a way to calculate how one quantity changes in response to changes in another. When you have a function, say \(y = f(x)\), the derivative \(f'(x)\) tells you the slope of the tangent line to the curve at any point \(x\). This slope indicates how steep the curve is at that particular point.

Derivatives can be used to find various properties of functions, like increasing or decreasing intervals, and can help in optimizing problems. In this exercise, we calculated the derivative of the function \(y = x^3 + 3x^2 + 3x - 10\) to find out how the curve behaves. This involves:

• Taking the derivative of each term separately, thanks to rules like the power rule: \((x^n)' = nx^{n-1}\).
• Applying the sum rule \((f + g)' = f' + g'\), where each part of your function is differentiated independently.

Using these rules, we differentiated the given function to get \(y' = 3x^2 + 6x + 3\). This new function \(y'\) gives us the slope of the original function at any given \(x\).

The slope of the tangent to a curve at a given point represents how steep the curve is at that point. Imagine a tangent line as just touching the curve at a single point without crossing it. The slope of this tangent line is calculated using the derivative we found earlier.

For any point \(x\) on the curve, the derivative \(f'(x)\) directly provides this slope. In this exercise concerning the function \(y = x^3 + 3x^2 + 3x - 10\), we needed the tangent slope at \(x = 2\). To find this, we simply evaluate the derivative at \(x = 2\):

• Plug the value into the derivative: \(y'(2) = 3(2)^2 + 6(2) + 3\).
• Solve to find \(y'(2) = 21\).

Thus, the slope of the tangent line to the curve at \(x = 2\) is 21, meaning it's quite steep at this point.

Differentiation is the process of finding a derivative, and it's a primary operation in calculus. This magical process helps us determine how functions change and is crucial to solving many real-world problems.

In the example given, we dealt with a polynomial function. Differentiation involves multiple steps and rules, primarily the power rule, which was used here extensively:

• For any term of the form \(x^n\), the derivative is \(nx^{n-1}\).
• For constants or terms without an \(x\), the derivative is zero.

By applying these rules,

• We differentiated \(y = x^3 + 3x^2 + 3x -10\) to get \(y' = 3x^2 + 6x + 3\).

Differentiation here helped us understand how the function's slope (its rate of change) behaves relative to \(x\). It's like peeling back layers to see the core behavior of a mathematical function.