Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Find the derivative of the function.$ f(t) = \sin^2 (e^{\sin^2 t}) $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

01:07

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Oregon State University

Baylor University

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:24

Find the derivative of the…

05:28

01:18

02:05

01:21

02:25

Differentiate the function…

01:15

01:03

00:38

0:00

Find the derivative.$$…

01:01

01:13

All right. So we're going to use the chain rule to find the derivative of this function, and this is going to test our chain rule to the limits here. First, I'm going to rewrite the function. So when you see sine squared the square dis really on the very most outside of it. So let's rewrite this as the sign of E to the it's get that e a little bit better e to the power of sine of t squared. So I moved that squared symbol as well, and then I'm going to move the other squared symbol to the very outside. So let's check this out. We have many, many layers to go here. Okay, so we start finding are derivative by taking the derivative of the outermost function, the squaring function. So we bring down the to and we raised everything inside it to the power one. Let's see if I have enough parentheses there of it. Now we move on to the next layer, which is sign, and we take its derivative, which is co sign so coastline of everything inside it there didn't need as many parentheses that time. Now we move on to the next layer, which is e to a power. And we do that as each of that power times the derivative of the power. So now, for the derivative of the power that has two layers, we have the squared on the outside. So we bring down the two and raise signed T to the first. And then we do the derivative of sign, which is co sign. Now we take a look and we ask ourselves, Is there anything we can do to make this simpler? Do we have any factors that are the same as other factors that we could combine? And the answer is no. But what we could do is take the two times the two and make that a four. And then any time we had a square that wasn't written in its proper position, we could put it back into its proper position. So we have four times the sign of e to the sine squared t times a co sign, uh, v to the signs where t times e to the sine squared T time scientist E times ko 70. Now you could have these terms in a variety of orders. It won't matter, like if you look up the answer in the book and they have all the same factors, but they're in a different order. It's okay. If they're factors they could be rearranged.

View More Answers From This Book

Find Another Textbook

01:48

A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If you we…

01:34

A juggler is performing her act using several balls. She throws the balls up…

00:57

Mikel is determining if the two triangles below could be similar based on th…

00:58

00:48

The product of two consecutive integers is 420. An equation is written in st…

02:47

Wilma and Greg were trying to solve the quadratic equation[x^2 + bx + c …

00:54

If a laptop originally costs $800, the balance due after the 10% discount an…

Paul had a job during his summer vacation. He earned 8.75 per hour. He worke…

00:59

Tobias’s closet has 1 red hat and 1 black hat; 1 white shirt, 1 black shirt,…