(a) Write |x|=√(x^2) and use the Chain Rule to show that...

Key Concepts

Composite Functions

Composite functions are functions formed by combining two or more functions, where the output of one function becomes the input of another. Differentiating composite functions requires careful application of the chain rule, and understanding their behavior often involves considering the individual functions and how they interact, especially when one of the functions is non-smooth or has discontinuities in its derivative.

Points of Non-Differentiability

Points of non-differentiability are specific values in the domain of a function where the derivative does not exist, often due to abrupt changes in the function's behavior such as sharp corners or cusps. In the context of functions involving absolute values, these points typically occur where the expression inside the absolute value changes sign.

Differentiation of the Absolute Value Function

Differentiating the absolute value function |x| involves recognizing that it can be expressed in terms of other functions (such as the square root of x²) and then applying differentiation rules appropriately, including the chain rule. The derivative of |x| is typically expressed as x/|x| for x ? 0, reflecting a difference in behavior on either side of zero.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is essential when differentiating functions that can be seen as the composition of two or more functions.

Transcript

00:01 So this is actually a little trick that i often use when you have absolute values functions.

00:07 And it's a nice way of figuring out what the derivatives are if you have an absolute value in an equation.

00:15 So the absolute value of x can be written as the square root of x squared because we take the positive by default.

00:24 By definition, we take the positive root.

00:27 So we can use the chain rule by setting u equal to x squared.

00:31 So then y equals square root of u.

00:35 And so dydx equals dy to u, d udx.

00:38 D y to u is 1 over 2 times the square of u.

00:43 Plugging u back in, we have 1 over 2 times the square of x squared, and then d udx is just 2x.

00:51 So 2's cancel out, and we get x divided by the square root of x squared, and that's just x divided by the absolute value of x.

00:59 And so what is this function? well, it's either 1 or minus 1.

01:04 It's 1 when x is greater than 0, and it's minus 1 when x is less than 0.

01:10 And when x equals 0, it's undefined.

01:13 So it has a kink in it like this, right? because if we plotted that, the value of x, it's just a v.

01:20 It's symmetric.

01:22 So the derivative is minus 1 over here and positive 1 over here, and that 0 it's not defined.

01:29 So they ask us to then use this kind of, i assume, use this kind of methodology to find the derivative of the absolute value of sine of x, which we can write as the square root of the sine squared of x.

01:45 And now u is sine squared of x.

01:47 And so, the y, du is 1 over 2 times the square of u, which is sine of x, sine square of x.

01:56 Then the u d x is 2 sine of x cosine of x, using a chain rule again.

02:05 And so after some simplification, we see the two is cancel out again.

02:09 This is absolute value of sine of x.

02:12 So we get cosine of x times the absolute value, this times cosine of x divided by absolute value of x.

02:25 And i plotted these somewhere.

02:28 Well, no.

02:30 How did i plot these? i know i plotted these, so i remember doing it.

02:35 And i see them, oh, let's see here, plot 20 and 23.

02:42 See, here we can find those.

02:45 20 and 23.