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Find the derivative of the function.$ f(x) = \sqrt{5x + 1} $

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00:22

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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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let's find the derivative of this function. And since it's a composite function with one function inside another, we're going to use the chain rule, and I like to rewrite it when I have a square root. I like to write it as five x plus one to the 1/2 power and use the power rule. So we start the chain ruled by taking the derivative of the, uh, the outside function. So we get 1/2 times five x plus one to the negative 1/2. Okay, we bring down the old power and we subtract one to get the new power. Now we multiply by the derivative of the inside, and that would be the derivative of five X plus one. And that would just be five. So we have our derivative, and now we're going to simplify. So what we can do is multiply the five and 1/2 and that's five halves. We can leave the five on the top and put the two in the denominator. And then because we have a negative exponents, we're gonna move that term to the denominator as well. So we have five x plus one to the 1/2 in our denominator. Now let's go ahead and change our notation back to radical notation. So we have 5/2 times a square root of five x plus one.

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