Prove, starting from the definition (and drawing a picture...

Key Concepts

Linearity of Differentiation

Differentiation is a linear operation, meaning the derivative of the sum of functions is the sum of their derivatives, and the derivative of a constant times a function is the constant times the derivative of the function. This property allows for the straightforward handling of both addition of a constant and multiplication by a constant when differentiating a function.

Derivative of Constant Functions

A constant function is one that does not change as the input changes; its graph is a horizontal line. Because there is no change (or slope) in a horizontal line, the derivative of a constant is zero. This principle is key when analyzing functions that have constant terms added or subtracted.

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the change in the variable approaches zero. This definition captures the idea of the instantaneous rate of change of the function, and geometrically, it represents the slope of the tangent line to the curve at that point.

Constant Multiple Rule

This rule states that if a function is multiplied by a constant, then its derivative is the constant multiplied by the derivative of the function. This follows directly from the limit definition of the derivative and is an expression of the linearity of the differentiation operator.

Transcript

00:01 This problem, they ask us to show some things about when you shift the function by a constant and also multiply it by a constant.

00:11 So if we shift the function by a constant, they want to show, we want to show that the derivatives are same.

00:16 Shift the function up by a constant.

00:19 So g of x by definition, or g prime of x by definition, is this.

00:24 And we were told that g of x is a, f of x plus c.

00:29 So we can substitute that in, and then here.

00:32 And what we can see is the c's cancel out.

00:35 And so we just get this equals f prime of x.

00:38 And we shouldn't be surprised by that because the constant that you use to shift, you know, g of x equals f of x plus c, all it does is move the function up or down.

00:54 It doesn't change the slope.

00:56 And so they ask us to illustrate that.

00:59 And what you can do is just basically any function you have, you just shift it up.

01:03 It doesn't change the slope of that function at all.

01:06 It just changed its position in the plane.

01:10 And in fact, it's arbitrary kind of anyway, because wherever you set, whenever you set y -equal to zero, right? you know, that's not, you know, in the grand scheme of things, where is y -equas zero? nobody knows.

01:24 So, you know, all you're doing is basically saying if you're shifting it, right? so, again, a shift in the y -direction.

01:35 Doesn't change the slope.

01:37 And actually we'll see, and the next problem, a shift in the x direction also doesn't change the slope...

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