Derivatives: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
Let $f(x)$ be a real function such that $f:R\rightarrow R$. The derivative $f\prime (x)$ of $f(x)$ is given by the formula:
Problem 2
Find the derivative of $f(x)=x^3-12x$, using the formula above.
Problem 3
Find the derivative of $f(x)=\sqrt{x+4}$ using the formula from question 1.
Problem 4
What is the equation of the line tangent to the graph of
$f(x)=x^{2}+3x$ at the point $(-2,2)$. Graph it.
Problem 5
What is the equation of the tangent to the graph of

$f(x)=x^{2}+3x$ at $(-2,2)$.
Problem 6
What is the equation of the line tangent to the graph of $f(x)=x^{3}+2$
that is parallel to line $3x-y-4=0$.
Problem 7
What is the value of $f\prime (c)$ if $f(x)=x^{3}+2x^{2}-1$ and $c=-2 $
Problem 8
What is the derivative of the function
$f(x)=\left\{ \begin{array}{c} x\quad \qquad x\leq 1 \\ x^{2}\quad \ \ \quad x>1 \end{array}\right\} $   at $x=0?$
Problem 9
What is the derivative of the function
$f(x)=\left\{ \begin{array}{c} x^{2}+1\quad \qquad x\leq 2 \\ 4x+3\quad \ \ \quad x>2 \end{array} \right\} $
at $x=3?$
Problem 10
What is the derivative of the function $f(t)=t^{3}+5t^{2}-3t+8$,
using derivative rules?

Problem 11
Consider the following functions and their derivatives:

$(i)\left\{ \begin{array}{c} f(x)=x^{2}+1\quad \\ f'(x)=2x\quad \ \end{array}\right\}$

$(ii)\left\{ \begin{array}{c} g(x)=\frac{x-1}{1-x}\quad \\ g'(x)=1\quad \ \end{array}\right\}$

$(iii)\left\{ \begin{array}{c} h(x)=\sqrt{x^{3}-x}\quad \\ h'(x)=\frac{3x^{2}-1}{2\sqrt{x^{3}-x}} \end{array} \right\} $
Problem 12
Consider the following functions and their derivatives:

$(i)\left\{ \begin{array}{c} f(x)=\sin (x^{3}-x+1)\quad \\ f'(x)=\cos (x^{3}-x+1)\quad \ \end{array}% \right\} \qquad (ii)\left\{ \begin{array}{c} g(x)=e^{x^{3}+x^{2}}\quad \\ g'(x)=\left( x^{3}+x^{2}\right) e^{x^{3}+x^{2}}\quad \ \end{array}\right\} \qquad $ $ (iii)\qquad \left\{ \begin{array}{c} h(x)=\sqrt{\ln (x^{3}+1)}\quad \\ h'(x)=\frac{x^{3}+1}{2\sqrt{\ln (x^{3}+1)}}\quad \ \end{array} \right\} $
Problem 13
$f(x)=\frac{1}{x}-\sin (x^{2})$ and $g(x)=x^{2}+\cos (2x+1)$
Find $\frac{d}{dx}(f(x)+g(x)) =$
Problem 14
Given $f(x)=e^{x^{2}+x}$ and $g(x)=\sqrt{x^{2}-x+1}$
Find $\frac{d}{dx}(f(x)\cdot g(x))=$ ?
Problem 15
If $f(x)=\sqrt{2x^{3}-3x^{2}+5}$ and $g(x)=\ln (x^{2}-x+1)$,
find $\frac{d}{dx}(\frac{f(x)}{g(x)})=$
Problem 16
Find the value of $k$ such that the line $y=5x-4$ is tangent to the graph of the function $f(x)=x^{2}-kx$
Problem 17
Find the value of $k$ such that the line $y=4x-1$ is tangent to the graph of the function $f(x)=kx^{4}$
Problem 18
Graph $y=x^{2}$ and $y=-x^{2}+5$

Graph two tangent lines to $y=-x^{2}+5$ through the points of intersection of the both functions.
What are the equations of the tangents?
Problem 19
Let the function of the vertical position of an object in free fall be $s(t)=-16t^{2}+v_{0}t+s_{0}$ where $v_{0}:$ is initial object speed $s_{0}:$ initial position of the object.

If a coin is tossed from the top of a building that has a height of $1362ft$

a) Determine the functions for position and speed of the coin.
b) Calculate the average speed in the range $\left[ 1,2\right]$.
c) Find the instant speed when $t=1,t=2$
d) Calculate the time it takes for the coin to reach the floor.
e) Determine the speed when the coin reach the floor

Solve it by yourself and check the solution.
Problem 20
From a height of $200ft$ a ball is thrown down with initial speed of $-22ft/\sec $, if the function of the position is $s(t)=-16t^{2}+v_{0}t+s_{0}$ what is the velocity $V_{1}$ in third second?

What is the velocity $V_{2}$ after descending $108$ft?
Problem 21
A projectile is launched upwards from the earth's surface, with an initial velocity of $120m/s$, if the position function of the projectile is $s(t)=-4.9t^{2}+V_{0}t+s_{0}$, what is the velocity $V_{5}$ in $5$th second, and $V_{10}$ in $10$th second.
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