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### Course: AP®︎/College Calculus AB>Unit 2

Lesson 7: Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule

# Tangents of polynomials

Let's explore the process of finding the equation of the tangent line to the graph of f(x) = x³ - 6x² + x - 5 at x = 1. By calculating the derivative and evaluating it at x = 1, we determine the slope and use point-slope form to obtain the tangent line equation.

## Want to join the conversation?

• The function here is cubic. The derivative is quadratic. I don't understand why evaluating f'(1) gets us the slope of the tangent line at 1? F' in this case isn't a line?!
• Derivatives don't have to be linear to still give us the slope of the tangent line. The point is that the derivative is a function that returns a single value at any point, which represents the slope of the tangent. The reason this works is shown in the proof videos - i.e., the ones showing the derivative expressed as the limit of a secant slope.

Consider this: if the slope of a function `f'` is changing in a way that is non-linear, how could you expect to find a linear function `f'` that tells you the slope of `f` at any point? By definition, that's not possible. So the derivative of a function whose slope changes nonlinearly will itself be nonlinear (by definition).

• At the constant x's power one meant the derivative was one, but I remember in one of the first videos covered that the derivative of any constant is zero. What made it one here?
• The derivative of a constant is 0, but x is not a constant. It's a variable. So we apply power rule, which works for functions of the form x^n.
• @　Why does the derivative of a constant become ZERO?

Is it becaue the slope of y = -5 is zero?
• Yes, that is the idea. More generally the derivative of any constant c is 0, because the slope of y = c is zero.

Have a blessed, wonderful day!
• What exactly is the derivative? How does it mathematically relate to solving the tangent. Also, do we use the derivative to solve anything else?
• The derivative is the rate of change so dy/dx is an infinitesimal change of y over an infinitesimal change in x. I would suggest you take look at differentiation by first principles to fully understand it.

So let say there is a car and you have want instantaneous speed of the car at 2.5 seconds. To find the speed of car we need to find the rate of change based on x and t where t is time and x is the distance travelled. So the average speed capital delta x / capital delta t s in physics as the capital delta is just difference in the distance and time so it (x_1 - x)/ (t_1-t). However we need to find the instantaneous speed. However the speed will be the same if we find the speed right after 2.5 seconds as in find the speed at 2.5^+ seconds so that infinitesmal difference in the times. Finding the average speed then would give the exact speed if you were try find the speed between the two points

So we can say the instantaneous speed of the car when t=2.5 seconds is

lim_dt->0 ( f(2.5+ dt) - f(2.5) ) / (2.5+dt -2.5) =

(f(2.5 + dt) -f(2.5)) / dt
• For finding values of x where the tangent line to the graph is horizontal using the equation 2x^2 + 3x - 1 would you simply find the derivative of the equation and set that answer to 0? And if so, the answer would be -3/4?
• That's correct. The statement "the derivative of 2x^2+3X-1 is 0" implies "x= -3/4".
• How would you find the derivative if there are fractions? For example: 3/x^2 -4/x^3?
I appreciate for the help.
• For the ones you gave, it's important to remember that 1/(x^a) = x^(-1). For the the derivative of 3/(x^2), you'd treat it as 3x^(-2) and use the power rule to get -6x^(-3). The rule also applies for fractional exponents like x^(3/2); it's derivative is (3/2)x^(1/2).
• When Sal took the derivative, he ended up with a quadratic polynomial. Why can we just plug in a number in that quadratic polynomial and come up with a slope? I wish he went into more detail that. Thanks
• Yes, you can just input the number but Sal just wanted to show another way to solve this question