Calculus 1

For the following problem, find the derivative of $f(x) = 5^{x^{3} - 4}$

Use the properties of logarithms to show that the derivative of $\log_{a}x = \frac{1}{(\ln{a})x}$

Use implicit differentiation to show that the derivative of $\ln{x} = \frac{1}{x}$ for $x > 0$

Note that many classes introduce logarithmic differentiation before implicit differentiation.

Find the derivative of $f(x) = \frac{\ln{(2x)}}{x^4}$

Find the derivative of $y = {(2x - 5)}^2$

Find the derivative of $y = {(x^2 + 3x)}^7$

Find the derivative of the following function

$y = \frac{\ln{(x - 1)}}{\sqrt{x^{\pi} + 1}}$

Find the derivative of $y$ with respect to $x$ for the following equation

$y(x+4) = x^2 - 3$

Find $\frac{dy}{dx}$ for $x^2 + y^3 = \log{(x + y)}$

Find the derivative, $y^{\prime}$ of the following implicit function

$y = \tan^{-1}(xy)$

Find $\frac{dy}{dx}$ and the slope of the tangent line at (0,3) for the curve given by

$y^3 + x^{2}y^{5} - x^4 = 27$

Show that for $y = \cos^{-1}(x)$ the first derivative, $\frac{dy}{dx} = \frac{1}{x^2 + 1}$

For the following function, find the first derivative

$\theta = \frac{\tan^{-1}(2r)}{\pi{r}}$

Find the derivative of the following hyperbolic function

$f(x) = \sin{(\sinh{(x)})}$

A 20 foot ladder is leaning against a wall and the base of the ladder is sliding away at 6 feet a second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 12 feet from the wall?

Let $y = 2(x^2 - 3x)$

a. Find $\frac{dy}{dt}$ when $x = 3$ given $\frac{dx}{dt} = 2$

b. Find $\frac{dx}{dt}$ when $x = 1$ given $\frac{dy}{dt} = 5$

A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

For the following function, use logarithmic differentiation to find $\frac{dy}{dx}$

$y = \frac{{(x + 2)}^2}{\sqrt{x^2 + 1}}$

Use logarithmic differentiation to find the first derivative of the function

$y = x^{x + 1}$

Find the derivative of $y = x^{\sin{(x)}}$