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If $ h(x) = \sqrt {4 + 3f(x)}, $ where $ f(1) = 7 $ and $ f'(1) = 4, $ find $ h'(1). $

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00:38

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

02:25

If $ h(x) = \sqrt {4 + 3f(…

02:34

If $$h(x)=\sqrt{4+3 f(x)},…

01:45

02:27

If $h(x)=[f(x)]^{2}+\sqrt{…

00:57

If $f(x)=4 x, g(x)=2 x-1,$…

01:09

00:55

Find $ f \circ g \circ h $…

00:39

If $f(x)=4 x-7,$ find $f(a…

00:41

all right. H of X is a composite function, and we want to find aged prime of one. So let's start by finding h prime of X. And to do that, we need to use the chain rule, and I'm actually going to start by rewriting h of x as four plus three f of X to the 1/2 power. So now, using the chain rule, the derivative is 1/2 times four plus three f of X to the negative 1/2 power times the derivative of the inside and the derivative of 40 and the derivative of three F of X would be three f prime of X. Now let's evaluate that at X equals one. So age prime of one is going to be 1/2 times four plus three times f of one to the negative 1/2 power times three times f prime of one. So we need to substitute f of one, which is seven, and we need to substitute f prime of one, which is four. So we have aged prime of one equals 1/2 times four plus three times seven to the negative, 1/2 power times three times For now, it's simplify that four plus three times seven that would be 25 and three times for that would be 12 25 to the negative 1/2 power is one over the square root of 25 so that would be 1/5. So we have 1/2 times 1/5 times 12 and that gives us 6/5.

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