Question

3.3 basic differentiation rules

1. using the power rule and other basic derivative rules, find the following:

a. $\frac{d}{dx}(5x^{3}+\frac{2}{x^{3}}+picdot e^{2})$
b. $\frac{d}{dt}(7t^{3/2}-\frac{6}{t^{3/2}})$
c. $\frac{d}{dx}(\frac{4sqrt3{x^{2}}+2x^{2}+7}{x^{2}})$ (note: you shouldnt need to use the quotient rule here!)

1. when throwing a softball directly upward from a height of 5 ft with an initial velocity of 50 ft/sec, the height of the softball after t seconds is given by $y(t)=-16t^{2}+50t + 5$ (until the ball hits the ground).

a. find the velocity $y(t)$.
b. over what time interval is the ball going upward?
c. at what time does the ball reach its maximum height?
d. what is the maximum height reached by the ball?

1. determine all the points on the graph of $y = x^{3}+x^{2}-x + 1$ where the tangent is horizontal.

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=nax^{n - 1}$, and the derivative of a constant $C$ is $0$.

Step2: Differentiate part A

For $y = 5x^{3}+\frac{2}{x^{3}}+\pi e^{2}=5x^{3}+2x^{-3}+\pi e^{2}$, using the power - rule[SSE Completed, Client Connection Error][SSE onError error]

Answer:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=nax^{n - 1}$, and the derivative of a constant $C$ is $0$.

Step2: Differentiate part A

For $y = 5x^{3}+\frac{2}{x^{3}}+\pi e^{2}=5x^{3}+2x^{-3}+\pi e^{2}$, using the power - rule[SSE Completed, Client Connection Error][SSE onError error]