Find d^3 y / d x^3. y=sin(x^3)

Name: Problem 6, Chapter 2: Calculus
Uploaded: 2021-06-12T22:46:22-08:00
Duration: 5 min 1 s
Channel: Eric Sigman
Description: Find d^3 y / d x^3. y=sin(x^3)

Key Concepts

Higher Order Derivatives

Higher order derivatives refer to the process of finding derivatives beyond the first derivative, such as the second, third, or nth derivative. This concept involves repeatedly differentiating the function and often requires the application of differentiation rules, including the chain and product rules.

Trigonometric Differentiation

Trigonometric differentiation involves differentiating trigonometric functions such as sine, cosine, and tangent. Knowing the derivatives of these trigonometric functions, in combination with the chain rule when they appear with composite arguments, is essential for solving problems like finding the third derivative of sin(x^3).

Chain Rule

The chain rule is a fundamental technique in differentiation that is applied when differentiating a composite function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This concept is crucial for functions like sin(x^3), where sin is the outer function and x^3 is the inner function.

Composite Functions

A composite function is one in which one function is applied to the result of another function, such as f(g(x)). Recognizing composite functions is essential because it helps determine the appropriate application of the chain rule during differentiation.

Transcript

00:01 All righty, so we've been asked to find the third order derivative of this function right here.

00:05 So we're going to start by taking our first order derivative, our first order derivative, y prime, will be cosine of x cubed times.

00:17 We're going to use our chain rule.

00:20 The derivative of x cubed is 3x squared.

00:24 And that is our first derivative.

00:26 Our second order derivative, y double prime.

00:30 We're going to use the product rule here.

00:32 So the derivative of the first is negative sign x cubed times the derivative of the inside, 3x squared, times our second term, 3x squared again.

00:53 Then we're going to add our first term, cosine of x cubed, times the derivative of our second term, which will just be 6x.

01:09 So if i put this all together, we'll rewrite y double prime.

01:16 That'll come out to be negative sign x cubed, all multiplied by 9x to the 4 plus cosine x cubed times 6x.

01:37 And for our final order derivative, we're going to have to do two product rules.

01:44 That's a y triple prime.

01:46 We're going to start with our first round of product rule, so put a bracket here.

01:54 Negative cosine of the inside.

02:00 We use our chain rule.

02:02 Take the derivative of x cubed, which is 3x squared.

02:06 Then we will take our just second term as usual, 9x to the 4.

02:13 Then we will add, or i subtract since the sign is negative, our first term, times...

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