Chain Rule For Finding Derivatives

Summary

This video tutorial provides an in-depth exploration of the chain rule in calculus, focusing on how to differentiate composite functions effectively. It outlines the fundamental formula and walks through various examples, including polynomial and trigonometric functions, demonstrating the step-by-step application of both the chain rule and the power rule. By illustrating complex functions and their derivatives, the tutorial aims to enhance viewers' understanding of differentiation, ultimately making the concept more accessible and easier to apply in different scenarios.

Takeaways

• 😀 The chain rule is essential for differentiating composite functions, where one function is nested inside another.
• 😀 To apply the chain rule, first differentiate the outer function while keeping the inner function unchanged.
• 😀 Always multiply by the derivative of the inner function when using the chain rule.
• 😀 The general power rule formula combined with the chain rule allows for differentiation of functions like U^n, where U is a function of x.
• 😀 Practice examples include finding derivatives of polynomial, trigonometric, and logarithmic functions using the chain rule.
• 😀 When differentiating functions like 5x + 3 raised to the 4th power, first apply the power rule before finding the derivative of the inside function.
• 😀 Understanding the derivatives of basic functions, such as sine, cosine, tangent, and their compositions, is crucial for applying the chain rule correctly.
• 😀 For more complex derivatives, like those involving products or quotients, combining the product rule or quotient rule with the chain rule may be necessary.
• 😀 Always rewrite functions in a more manageable form when necessary, such as converting roots or fractions to powers, to simplify differentiation.
• 😀 The final answer should be fully simplified, combining like terms and expressing the result clearly.

Q & A

What is the chain rule in calculus?

The chain rule is a formula for computing the derivative of a composite function. It states that to differentiate a composite function f(g(x)), you first differentiate the outer function f while keeping the inner function g the same, then multiply by the derivative of the inner function g.

How do you apply the chain rule to the function (5x + 3)^4?

To apply the chain rule, first differentiate the outer function (which is the power of 4) to get 4(5x + 3)^3. Then, multiply by the derivative of the inner function (5x + 3), which is 5. Thus, the final derivative is 20(5x + 3)^3.

What is the general power rule formula combined with the chain rule?

The general power rule formula states that for a function u^n, the derivative is n * u^(n-1) * du/dx, where u is a function of x. The chain rule requires you to multiply by the derivative of the inner function u.

What is the derivative of sin(6x)?

The derivative of sin(6x) using the chain rule is 6cos(6x). You differentiate the outer function sin to get cos, keep the inner function (6x) the same, and then multiply by the derivative of the inner function (6).

How do you differentiate ln(x^7)?

To differentiate ln(x^7), you first rewrite it as 7ln(x) and then differentiate to get 7/x. This uses the chain rule where the derivative of ln(u) is 1/u multiplied by du/dx.

What is the process to differentiate a function like (x^2 - 7)^12?

Rewrite the function as (x^2 - 7)^12, apply the power rule to bring down the 12 to get 12(x^2 - 7)^11. Then, multiply by the derivative of the inside function (2x), resulting in the final derivative: 24x(x^2 - 7)^11.

What is the derivative of 1/(x^2 + 8)^3?

First, rewrite it as (x^2 + 8)^(-3). The derivative is -3(x^2 + 8)^(-4) multiplied by the derivative of the inside function (2x), giving -6x/(x^2 + 8)^4.

How do you find the derivative of composite trigonometric functions?

For composite trigonometric functions, differentiate the outer function first, then keep the inner function the same, and multiply by the derivative of the inner function. For example, to differentiate cos(tan(x^4)), you would get -sin(tan(x^4)) * sec^2(x^4) * 4x^3.

What rule do you use when differentiating the product of two functions?

When differentiating the product of two functions, you use the product rule. If you have functions f(x) and g(x), the derivative is f'(x)g(x) + f(x)g'(x).

How do you apply both the chain rule and quotient rule in differentiation?

To apply both the chain rule and quotient rule, first differentiate the outer function while applying the chain rule. For a quotient f/g, use the quotient rule: (g * f' - f * g')/g^2. Be sure to apply the chain rule where necessary for inner functions.