class: center, middle, inverse, title-slide # Power rule ## Differential Calculus ### Arturo Sánchez González ### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a> ### April 2021 --- # What is the velocity of the car? .pull-left[ <br/> Problem <br/> <span style="color:blue"> Let's suppouse that the distance traveled by a car is described by `\(s(t) = t^3\)` (with `\(t\geq 0\)`). <span/> <br/> <!-- - <span style="color:blue"> At time `\(t=0\)`, is the car moving? <span/> --> - <span style="color:blue"> What is the velocity of the car at `\(t = 2\)`? <span/> - <span style="color:blue"> Can you generalize the previous result to know the velocity for every time? <span/> ] .pull-right[ <br/> <center> <img src="images/senalamiento.png" width="250"> <center/> Modified image of the one available at (_Todas las señales de tránsito preventivas con imagen [2021]_, n. d.) ] --- # Derivative (formal definition, just for general knowledge) <br/ > <br/> The **derivative of f** with respect to `\(x\)` is `$$\frac{df}{dx}(x) = \lim\limits_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$$` this is called **Leibniz notation**. <br/> Other usual notation of the derivative of a function is the **Lagrange notation** given by `$$f'(x) = \lim\limits_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$$` (sometimes it is read as _f prime of_ `\(x\)`) --- # Solving the problem (1) .pull-left[ ## Main problem solving strategy 1. Recall that the derivative is interpreted as the instantaneous rate of change. 2. Recall that the _velocity_ at a given point can be considered as an _instanteneous rate of change_. 3. In view of the previous points, **calculate the derivative of the function at the given point** ] .pull-right[ ##Solution 1. The velocity of the car can be obtained as the derivative of the position function because it is an instantaneous rate of change. 2. Proceed to calculation. The velocity of the car at `\(t=2\)` is `$$\frac{ds}{dt}(2) = \lim\limits_{h\rightarrow 0}\frac{s(2+h) - s(2)}{h}$$` ] --- # Solving the problem (2) <br/> <br/> <br/> <br/> <br/> <br/> <center> <span style="font-size:35px"> What can be done to answer part 2 of the problem ? <span/> <center/> --- # Power rule <br/> <span style="font-size:40px"> If `\(f(x) = x^n\)`, then <span/> <br/> <span style="font-size:40px">$$\frac{df}{dx}(x) = n x^{n-1}$$ <span/> --- # Examples .pull-left[ ## Function `$$f(x) = x^2$$` <br/> <br/> `$$s(t) = t^3$$` ] .pull-right[ ## Derivative `$$f'(x) = 2x$$` `$$f'(5) = 2(5) = 10$$` <br/> `$$s'(t)= 3t^2$$` `$$s'(2) = 3(2)^2 = 3(4) = 12$$` ] --- # Stretching-out <span style="font-size:25px"> Find the derivatives of the following functions: <span/> - <span style="font-size:25px"> `\(f(x) = x^{25}\)` <span/> - <span style="font-size:25px"> `\(g(t) = t^{7}\)` <span/> - <span style="font-size:25px"> `\(h(x) = x^{100}\)` <span/> - <span style="font-size:25px"> `\(f(x) = x^{1}\)` <span/> - <span style="font-size:25px"> `\(l(t) = t^{2021}\)` <span/> --- # References <br/> - _Todas las señales de tránsito preventivas con imagen [2021]_. (n. d.). Libro vial [cultura general]. Retrieved April 19, 2021, from https://librovial.blogspot.com/2020/01/todas-las-senales-de-transito.html