Slope of a Tangent Line Find the slope of the tangent line to the graph of f at the given point.

step 1 So we're given the function F of X is equal to 6 divided by X plus 1. And what we want to do is

Slope of a Tangent Line Find the slope of the tangent line...

Key Concepts

Derivative

The derivative of a function gives the instantaneous rate of change and is defined as the limit of the difference quotient. It provides the slope of the tangent line to the function's graph at any point, encapsulating how the output value changes as the input changes incrementally.

Tangent Line

The tangent line to a curve at a specific point represents the best linear approximation of the function at that point. Its slope is determined by the derivative of the function at that point, meaning it shows how steeply the function is rising or falling at that location.

Differentiation of Rational Functions

Differentiating a rational function, which is a ratio of two polynomials, often involves using rules such as the power rule along with the chain rule or quotient rule. In many cases, a simplification step such as rewriting the function in exponent form makes differentiation more straightforward.

Transcript

00:01 So we're given the function f of x is equal to 6 divided by x plus 1.

00:06 And what we want to do is we want to find the slope of the tangent line to the graph of f at 2 comma 2.

00:12 That's where we want to find it.

00:13 And so how do we do it? well, basically that means we want to find f prime of 2, the derivative of the function at x equals 2.

00:22 And so we want to do this using the limit definition because that's how the chapter in the book wants us to do it.

00:27 And so, so to find the derivative of a function at a point using the limit definition, that's going to be the limit as x approaches a of f of x minus f of a divided by x minus a.

00:45 And so f prime of 2 is equal to the limit as x approaches 2 of f of x minus f of 2 divided by x minus 2.

00:56 And so now what i can do is i can scroll down and then we can do some substituting and all that.

01:07 So f prime of 2 is equal to the limit as x approaches to of well, our f of x is just our function.

01:15 So 6 over x plus 1 minus f of 2.

01:19 Well, if our f of x is 6 over x plus 1, then our f of 2 is going to be 6 over 3 or 2.

01:25 2.

01:26 So minus 2, i think i did it right.

01:29 Yeah, minus 2 divided by x minus 2.

01:32 And so f prime of 2 is equal to the limit as x approaches 2 of, we can simplify this, get common denominator.

01:43 So we're going to get 6 over x plus 1 minus 2 times x plus 1 divided by x plus 1 all over x minus 2...

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