(a) If F(x)=5 x /(1+x^2), find F^'(2) and use it to find an equation of the tangent line to the

(a) If F(x)=5 x /(1+x^2), find F^'(2) and use it to find an...

(a) If $F(x)=5 x /\left(1+x^{2}\right)$, find $F^{\prime}(2)$ and use it to find an equation of the tangent line to the curve $y=5 x /\left(1+x^{2}\right)$ at the point (2,2)
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Key Concepts

Tangent Line Equation

The tangent line to a curve at a specific point is the straight line that just touches the curve at that point, having the same slope as the function at that point. The equation of the tangent line is typically derived using the point-slope form, where the slope is given by the derivative evaluated at the point and the point coordinates are those on the function.

Graphical Representation

Graphical representation involves plotting both the function and its tangent line on the same coordinate system to visually illustrate how the tangent approximates the function at the point of tangency. This depiction emphasizes the concept of local linear approximation and aids in understanding the behavior of the function near the specific point.

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to change in the input variable. It is a fundamental concept in calculus that helps determine the slope of the function at any given point, and is foundational for understanding instantaneous rates of change and for modeling change in various fields.

Quotient Rule

The quotient rule is a differentiation technique used for finding the derivative of a function that is the ratio of two differentiable functions. It provides a systematic approach to differentiate functions written in the form f(x)/g(x), by accounting for the derivatives of both the numerator and the denominator.

Transcript

00:01 Today we'll be finding the equation of a line change into the curve y equals 5x over 1 plus x squared at the point 2 .2 to do this we'll start by finding the slope, which means we have to find the derivative.

00:12 So y prime equals with the quotient rule, as i learned it we have the bottom 1 plus x squared times the derivative of the top which is 5 minus the top which is 5x times the derivative of the bottom which is 2x all divided by the bottom of the bottom squared.

00:31 So 1 plus x squared squared.

00:35 So we'll plug two in for x since that is our x value, and we will have 1 plus 2 squared times 5 minus 5 times 2 times 2 times 2.

00:51 I'll rewrite this, times 2 all over 1 plus 2 squared squared.

01:01 So we'll simplify that.

01:04 That will be 5.

01:06 Times 5 minus 10 times 4 over 5 squared, which is 25 minus 40 over 25.

01:20 And 25 minus 40 is 15, well, negative 15.

01:25 So negative 15 over 25, which when you reduce it, gives us negative 3 over 5 as our slope.

01:33 So then, to find the equation of a line, it's y minus y 0 equals m, times x minus x 0 and x 0 y 0 we're given to us as 2 2.

01:47 So just to plug in, we have y minus 2 equals negative 3 5ths times x minus 2.

02:01 We'll explain that and solve for y.

02:04 So y equals negative 3 5ths plus 6, 10 5ths...

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