Basic Rules of Differentiation: Videos & Practice Problems

Evaluate the following derivative using the graph given below.

$\frac{\text{d}}{\text{dx}}{(8f\left(x\right)+12g\left(x\right))}_{x=2}$

Evaluate the following derivative using the graph given below.

${\left(f^{-1}\right)}^{\prime }\left(4\right)$

Identify the function $g$ and value of $a$ such that the following limit represents $g^{\prime }\left(a\right)$. Then, evaluate the limit by finding the value of $g^{\prime }\left(a\right)$.

$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}$

A small town's expansion is modeled by the function $S\left(t\right)=-\frac{1}{40}{t}^2+3t+15$, for $0 ≤ t ≤ 60$, where $S$ represents the area in square kilometers and $t$ is the number of years since $2000$. If the population density remains steady at $800$ people per square kilometer, what is the population growth rate in $2025$?

For the function $h\left(x\right)=5\sqrt{x}-3x$, identify all the points on the graph of $h$ where where the graph has a horizontal tangent.

The graphs of $f$ and $g$ are shown. Find $K^{\prime }\left(6\right)$ if $K=5f-2g$.

Consider the curve defined by $y=x^3-8x+5$. Determine the smallest slope of the curve and the point at which it occurs.

A car accelerates from rest along a straight road and its distance from the starting point is given by the equation $S=\frac{5}{4}{t}^2$, where $S$ is the distance in meters and $t$ is the time in seconds. If the car needs to reach a speed of $90\text{ km/h}$ to safely merge onto a highway, how long will it take to reach this speed, and what distance will the car cover?

Evaluate the following derivative using the table given below.

$\frac{\text{d}}{\text{dx}}{(5f\left(x\right)+4g\left(x\right))}_{x=6}$

Evaluate the following derivative using the table given below.

${\left(g^{-1}\right)}^{\prime }\left(17\right)$

If $h\left(x\right)=x^3$ is a function defined on all real numbers, what is the nature of $h^{\prime }\left(x\right)$?

Evaluate the following derivative using the graph given below.

${\left(f^{-1}\right)}^{\prime }\left(5\right)$

Evaluate the following derivative using the graph given below.

${\left(f^{-1}\right)}^{\prime }\left(f\left(3\right)\right)$

Evaluate the derivative of the function $g\left(x\right)=3x^{\frac{1}{3}}-2$ at $x=8$.

If $m^{\prime }\left(2\right)=7$ and $n^{\prime }\left(2\right)=-5$, find the value of ${(m+n)}^{\prime }\left(2\right)$.

Consider the function $g\left(x\right)=\mid x-2\mid$. Use the graph of $g\left(x\right)$ to find $g'(x)$ for $x\ne 2$.

Identify the function $g$ and value of $a$ such that the following limit represents $g^{\prime }\left(a\right)$.

$\underset{x\to 0}{\lim }\frac{e^{2x}-1}{x}$

Identify the function $g$ and value of $a$ such that the following limit represents $g^{\prime }\left(a\right)$. Then, evaluate the limit by finding the value of $g^{\prime }\left(a\right)$.

$\underset{x\to 0}{\lim }\frac{e^{2x}-1}{x}$

Find the derivative of the function $h\left(r\right)=r^8$.

Find the derivative of the function $h\left(y\right)=9y^5$.

Calculate the derivative of the function $g\left(x\right)=\frac{x^3}{3}+5$.

Determine the derivative of the function $p\left(t\right)=4t^8+3e^t-5$.

Calculate the derivative of the function $k\left(y\right)=7\sqrt{y}$.

Find the derivative of the function $h\left(x\right)=\frac{5\sqrt{x}}{6}$.

Identify the function $g$ and value of $a$ such that the following limit represents $g^{\prime }\left(a\right)$.

$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}$

Find the derivative of the function $f\left(x\right)=7x^6-\frac{7}{4}{x}^4+3x+10$.

Find the derivative of the function $g\left(y\right)=5\sqrt{y}-\frac{1}{5}{y}^5+y-3$.

Determine the derivative of the function $h\left(x\right)=5x^2-2$, and $h^{\prime }\left(c\right)$ for $c=-1$ and $c=2$ .

An apple falls from a tree, and the distance it travels is given by $y\left(t\right)=15t^2$, where $y$ is in meters and $t$ is in seconds. Determine $y^{\prime }\left(t\right)$. What are the units of the derivative, and what does it measure?

A pebble is dropped from a bridge, and the distance it falls is given by the equation $d\left(t\right)=15t^2$, where $d$ is in meters and $t$ is in seconds. If the pebble takes $7$ seconds to reach the water below, what is the height of the bridge? Also, calculate the pebble's speed upon hitting the water in kilometers per hour.