Find f^'(a). f(x)=√(1-2 x)

Name: Problem 31, Chapter 2: Calculus
Uploaded: 2021-08-26T00:25:22-08:00
Duration: 4 min 5 s
Channel: Leon Druch
Description: Find f^'(a). f(x)=√(1-2 x)

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Key Concepts

Derivative

The derivative is a fundamental concept in calculus that measures the instantaneous rate of change of a function with respect to its variable. It provides information on how the function behaves and changes at any given point.

Chain Rule

The chain rule is a formula used to differentiate composite functions, where a function is applied inside another function. It allows the differentiation process to be broken down into the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Differentiation of a Square Root Function

Differentiating a square root function involves recognizing it as a power function (expressed as an exponent of 1/2) and then applying the power rule in conjunction with the chain rule. This method is used to handle functions under a radical sign effectively.

Evaluation at a Specific Point

Evaluating a derivative at a specific point involves first finding the general formula for the derivative and then substituting the particular value of the variable into this formula to obtain the instantaneous rate of change at that point.

Transcript

00:01 Okay, in this function, f of x is equal to the square root of 1 minus 2x, and we want to find the derivative at a.

00:08 We are going to use the chain rule, which says the derivative of our function with respect to x, d, y, dx, equals the derivative of the function with respect to u times the derivative of u with respect to x.

00:34 U in this case is going to be the 1 minus 2x.

00:39 So we're taking the square root of 1 minus 2x or the square root of you.

00:43 So now i'm going to rewrite our function f of x as 1 minus 2x.

00:57 And instead of writing as the square root of 1 minus 2x, the square root is the same thing as raising to the 1 half power.

01:04 So, f of x can be rewritten as 1 minus 2x to the 1 half power.

01:18 Now, if this is f of x, we're now going to differentiate this function.

01:28 Now, if you think of this as the u, the derivative of u to the 1 half is 1⁄2 times u, which is this in parentheses, and then 1 half, you would have to subtract 1 from this exponential.

01:48 And see you would get one half minus one is negative one half.

01:53 So that's the derivative of u to the one -half.

02:00 It's one -half times you to the negative one -half.

02:03 But then it's a little hidden here.

02:06 We still have to do the derivative of you with respect to x.

02:10 Let me show you that...

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