Learn Derivatives with Practice Problems and Examples

An airplane flies at an altitude of 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation ${\displaystyle \theta }$ is changing when the angle is 
a) ${\displaystyle \theta ={30}^{\circ }}$ 
b) ${\displaystyle \theta ={75}^{\circ }}$

Calculate the directional derivative of ${\displaystyle f(x,y,z)=xy+yz+zx}$ at P(1, -1, 3) in the direction of Q(2, 4, 5).

Find the derivatives of ${\displaystyle f\left(x\right)=\frac{3}{2}{x}^4+2x^3-5x^2+9}$

Find the derivatives of ${\displaystyle h\left(x\right)=(-7(x^2-8)2)2x}$

Need help with the following question Compute the directional derivative of
${\displaystyle f(x,y)=e^{2}x-5y}$

Define ${\displaystyle g(x,y)=x^2+y^2-4xy+3y+2}$

Find the derivative ${\displaystyle \frac{d}{dx}\left[\arctan (x^2+1)\right]}$
a) ${\displaystyle {\left(arc\sec (x^2+1)\right)}^2}$
b) ${\displaystyle \frac{1}{2+x^2}}$
c) ${\displaystyle \frac{1}{1+{(x^2+1)}^2}}$
d) ${\displaystyle \frac{2x}{1+{(x^2+1)}^2}}$

Find the point on the line $y=4x+3$ that is closest to the origin.
(x,y)=( )

Find ${\displaystyle \frac{dy}{dx}}$ and ${\displaystyle \frac{d^{2}y}{{dx}^2}}$.
${\displaystyle x=e^t,y=t{e}^{-t}}$
For which values of t is the curve concave upward?

$y(x^2-1)dy+x(y^2+1)dx=0$

Check whether the given function is satisfying the Fubini’s theorem about ‎mixed partial derivative ‎
${\displaystyle f(x,y)=3x^2\sin \left(4xy\right)}$