47-50 Find y^' and y^'' y=cos(sin3 θ)

Name: Problem 47, Chapter 2: Calculus
Uploaded: 2021-03-18T19:24:27-08:00
Duration: 4 min 38 s
Channel: Jacquelyn Mouck
Description: 47-50 Find y^' and y^'' y=cos(sin3 θ)

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions, allowing us to differentiate a function by taking the derivative of the outer function evaluated at the inner function and then multiplying by the derivative of the inner function. This rule is essential in the differentiation of functions formed by the composition of trigonometric and other elementary functions.

Trigonometric Function Derivatives

Knowing the derivatives of trigonometric functions is critical since they form the basis of differentiating composite trigonometric expressions. For example, the derivative of cosine is negative sine and the derivative of sine is cosine. These standard derivatives are combined with the chain rule when differentiating functions involving nested trigonometric operations.

Second Derivative Computation

Computing the second derivative involves differentiating the first derivative. This process often requires the successive application of the chain rule and sometimes the product rule when the first derivative is a product of functions. Understanding how to systematically differentiate complex derivatives is important for analyzing the curvature and concavity of functions.

Product Rule

The product rule is used when differentiating a product of two or more functions. In the context of finding higher order derivatives of composite functions, the product rule is often necessary when the first derivative is expressed as a product of functions that each depend on the independent variable. It complements the chain rule in the computation of the second derivative.

Transcript

00:01 In this problem, we're going to find the first and second derivative of y.

00:05 And to do that, we're going to use the chain rule.

00:10 So first i'm going to find the first derivative of y, which can be written as y prime.

00:16 And in y, we have a function within a function.

00:23 So we have this outer function of cosine and this inner function of sine 3 theta.

00:27 So to find the derivative, we have to use the chain rule.

00:30 And the chain rule says that we can find the derivative by taking the derivative of outside times the derivative of the inside.

00:38 So first the derivative of the outside is the derivative of cosine, which is negative sign.

00:43 So we take negative sign of this whole thing.

00:49 And now we need to find the derivative of the inside term.

00:52 Well, sine 3 theta is again a function within a function.

00:55 We have the 3 theta within the sign function.

00:59 So we have to do the chain rule again.

01:02 We take the derivative of the outside.

01:03 So the derivative of sine is cosine 3 -theta.

01:09 And then again, we have to multiply by the derivative of the inside function.

01:12 So the derivative of 3 -theta is just 3.

01:17 Putting this all together, we can write this as negative 3 -cosine 3 -theta times sine of sine 3 -theta.

01:27 And that is our answer for the first derivative.

01:34 To find the second derivative, we're going to have to use a combination of the product rule and the chain rule.

01:39 So i'm going to look at my first derivative and split it into two products and then use the product rule to find the second derivative.

01:56 So remember the second derivative is the derivative of the first derivative...

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