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Find the value of $ c $ such that the line $ y = \frac{3}{2}x + 6 $ is tangent to the curve $ y = c \sqrt{x}.

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03:22

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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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it's lower. So when you read here So we're gonna start off by differentiating C square affects when we get C over two square root of X. This is for the line. Why equals three Half plus six is tangent to the curve, See skirt of X. So the slope off the line three over two should be equal to the derivative. So we see that this is equal to three over two. When we get our X value to be C square over nine, then we're going to solve the equation. So we get why is equal to three Have X Plus six we have. Why is equal to see square it of X. Don't we get three has X Plus six is equal to C square of X. Then we're going to simplify the equation by squaring both sides when we get nine over four Up Square plus 18 x was 36 is equal to see square necks, and we're going to simplify our equation even more. We got 9/4 c square over nine, which we got right here square plus 18 c square over nine plus 36 this equal to see square times C square over nine, and then we just simplify continuously. You see that? It's only positive. Um, Val, You see, since the only positive value sea gives a curve that intersects the line.

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