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Find an equation of the tangent line to the curve at the given point.$ y = \sqrt {1 + x^3}, (2, 3) $

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$y=2 x-1$

00:55

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Oregon State University

University of Nottingham

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.

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Find an equation of the t…

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we're going to find the equation of the tangent line to the curve y equals square root one plus X cubed at the 10.23 The slope of the tangent line will be the derivative at that point. So let's start by finding the derivative. And I'm going to rewrite why, as one plus X cubed to the 1/2 power. And so then I can use the chain rule, starting with the derivative of the 1/2 power function. Bring down the 1/2 and raise one plus X cubed to the negative 1/2 power. And then we multiply that by the derivative of the inside, the derivative of one plus x cubed is three x squared. Now we're evaluating this derivative at X equals two. We could simplify first, or we could substitute the two in first and simplify after. So that's what I'm going to do. So 1/2 times one plus two cubed to the negative 1/2 times three times two squared. Okay, so this gives us 1/2 times nine to the negative 1/2 times 12 and nine to the negative. 1/2 is 1/3 so we have 1/2 times 1/3 times 12 So that is just too. So are slope is, too. Now we can use point slope form. Why minus y one equals m times X minus X one. To find the equation of the line where the point x one y one is 23 So we have why minus three equals air slope to times the quantity X minus two and we can simplify that and we'll have the equation of the line. So let's distribute the two. So why minus three equals two X minus four and then at three to both sides and we have y equals two x minus one.

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