The Fundamental Differentiation Rules: Understanding Chain, Product, and Quotient

In calculus, differentiation involves several rules that serve different purposes. In this article, we will delve into the three main rules you should know: the chain rule, product rule, and quotient rule.

Chain Rule

The chain rule is used when differentiating a composite function, which is a function within another function. The formula for this rule is dy/dx = (dy/du) X (du/dx), where y is a function of u and u is a function of x. It is important to note that not all formulas may be provided in your course material, so it's beneficial to memorize them for exams.

To apply the chain rule, first, rewrite the function in terms of y and u. Then, find the derivatives of y and u separately. Next, multiply them together and express the final answer in terms of x.

For example:

If y = sin(u) and u = 2x + 1, find dy/dx.

• Identify y and u: y = sin(u) and u = 2x + 1
• Find dy/du and du/dx: dy/du = cos(u) and du/dx = 2
• Substitute them into the formula: dy/dx = (cos(u)) X (2)
• Express the answer in terms of x: dy/dx = 2cos(2x + 1)

Product Rule

The product rule is used when differentiating the product of two functions. Like the chain rule, it's crucial to ensure that the functions are multiplied together and not nested within each other. The formula for this rule is dy/dx = u dv/dx + v du/dx, where y = uv and u and v are functions of x.

Using this rule, first, identify u and v. Next, differentiate u and v separately. Then, plug them into the formula and solve for dy/dx.

For example:

If y = x^2 and v = 3sin(x), find dy/dx.

• Identify u and v: u = x^2 and v = 3sin(x)
• Differentiate u and v: du/dx = 2x and dv/dx = 3cos(x)
• Substitute them into the formula: dy/dx = (x^2) X (3cos(x)) + (3sin(x)) X (2x)
• Simplify and express the answer in terms of x: dy/dx = 6xcos(x) + 2x^2sin(x)

Quotient Rule

The quotient rule is used when differentiating the quotient of two functions, meaning one function is divided by the other. The formula for this rule is dy/dx = (v du/dx - u dv/dx)/v^2, where y = u/v and u and v are functions of x.

To use this rule, identify u and v and differentiate them separately. Then, plug them into the formula and solve for dy/dx.

For example:

If y = 2x^3 and v = cos(x), find dy/dx.

• Identify u and v: u = 2x^3 and v = cos(x)
• Differentiate u and v: du/dx = 6x^2 and dv/dx = -sin(x)
• Substitute them into the formula: dy/dx = ((cos(x)) X (6x^2) - (2x^3) X (-sin(x)))/(cos(x))^2
• Simplify and express the answer in terms of x: dy/dx = (6x^2cos(x) + 2x^3sin(x))/(cos(x))^2

Key Takeaways

• The three main differentiation rules are the chain rule, product rule, and quotient rule.
• Each rule has a specific purpose and formula to be used.
• The chain rule is used for composite functions, the product rule is used for products of two functions, and the quotient rule is used for quotients of two functions.

In summary, comprehending and utilizing these differentiation rules is vital in solving advanced calculus problems. Practice and commit these rules to memory to excel in your exams!