Calculus is a branch of mathematics that deals with the study of change and motion. One important concept in calculus is the derivative, which measures how a function changes with respect to its inputs. In this article, we will be calculating the derivatives of three different functions using the rules of calculus.

1. Function f(x) = 3x^2 – 2x + 5:
To find the derivative of f(x), we can apply the power rule and the constant rule. The power rule states that the derivative of x^n is nx^(n-1), and the constant rule states that the derivative of a constant is zero. Therefore, the derivative of f(x) is:

f'(x) = d/dx (3x^2) – d/dx (2x) + d/dx (5)
= 6x – 2

2. Function g(x) = (2x^3 + 4x^2 – 3x) / (x – 1):
To find the derivative of g(x), we can use the quotient rule. The quotient rule states that the derivative of (f(x) / g(x)) is (f'(x)g(x) – f(x)g'(x)) / (g(x))^2. Therefore, the derivative of g(x) is:

g'(x) = (d/dx (2x^3 + 4x^2 – 3x) * (x – 1) – (2x^3 + 4x^2 – 3x) * d/dx (x – 1)) / (x – 1)^2
= (6x^2 + 8x – 3) * (x – 1) – (2x^3 + 4x^2 – 3x) / (x – 1)^2

3. Function h(x) = sqrt(x^2 + 1):
To find the derivative of h(x), we can use the chain rule. The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In this case, f(x) = sqrt(x), and g(x) = x^2 + 1. Therefore, the derivative of h(x) is:

h'(x) = d/dx [sqrt(x^2 + 1)]
= (1/2) * (x^2 + 1)^(-1/2) * 2x
= x / sqrt(x^2 + 1)

In conclusion, to calculate the derivatives of the given functions, we apply the power, constant, quotient, and chain rules of calculus. These rules allow us to find the rate of change or slope of a function at any point, which is an essential concept in calculus.
微積分是數學的一個分支，用於研究變化與運動。微積分中一個重要的概念是導數，它測量函數相對於其自變量的變化。在本文中，我們將應用微積分的規則計算三個不同函數的導數。

1. 函數 f(x) = 3x^2 – 2x + 5：
要找到 f(x) 的導數，我們可以應用冪規則和常數規則。冪規則指出 x^n 的導數是 nx^(n-1)，常數規則則指出常數的導數為零。因此，f(x) 的導數為：

f'(x) = d/dx (3x^2) – d/dx (2x) + d/dx (5)
= 6x – 2

2. 函數 g(x) = (2x^3 + 4x^2 – 3x) / (x – 1)：
要找到 g(x) 的導數，我們可以使用商規則。商規則指出 (f(x) / g(x)) 的導數是 (f'(x)g(x) – f(x)g'(x)) / (g(x))^2。因此，g(x) 的導數為：

g'(x) = (d/dx (2x^3 + 4x^2 – 3x) * (x – 1) – (2x^3 + 4x^2 – 3x) * d/dx (x – 1)) / (x – 1)^2
= (6x^2 + 8x – 3) * (x – 1) – (2x^3 + 4x^2 – 3x) / (x – 1)^2

3. 函數 h(x) = sqrt(x^2 + 1)：
要找到 h(x) 的導數，我們可以使用鏈式法則。鏈式法則指出 f(g(x)) 的導數是 f'(g(x)) * g'(x)。在這個例子中，f(x) = sqrt(x)，g(x) = x^2 + 1。因此，h(x) 的導數為：

h'(x) = d/dx [sqrt(x^2 + 1)]
= (1/2) * (x^2 + 1)^(-1/2) * 2x
= x / sqrt(x^2 + 1)

總結而言，要計算所給函數的導數，我們應用微積分的冪、常數、商和鏈式規則。這些規則使我們能夠找到函數在任意點的變化率或斜率，這是微積分中的一個重要概念。