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## AP®︎/College Calculus BC

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

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

# Tangents of polynomials

Sal finds the equation of the line tangent to the graph of f(x)=x³-6x²+x-5 at x=1.

## 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).

I hope that makes sense and somewhat answers your question.
• 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!
• 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).
• 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
• Why didn't Sal just use point-slope form there at the end? You find a point and a slope (or derivative), so it seems much more intuitive to me to do it that way instead of plugging the pieces into the form y = mx +b. I guess you could do that. But on the AP exam, point-slope form is a perfectly acceptable form and there might even be multiple choice questions in that format.
• It doesn't matter which form is used as both will describe the same line. Personally, I prefer slope-intercept form as it seems far mare intuitive to me than point-slope form in any case.
• 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".