Implicit differentiation and related rates

Practice sheet

Fresh questions from the whole chapter, to do on paper.

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$x^2 + 6y^2 = 63$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ .
$xy + 7x = 19$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ .
$y^2 = x^3 + 5x$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ .
Find the slope of the tangent line to the circle $x^2 + y^2 = 25$ at the point $(4, 3)$ .
Write the tangent line to $x^2 + y^2 = 20$ at $(4, 2)$ : enter the right-hand side of $y = \ldots$
The point $(1, 2)$ is on the curve $x^2 + xy + y^2 = 7$ . Find the slope of the tangent line there.
$f(x) = x^3 + 8x$ has an inverse $g$ . Note $f(1) = 9$ . Find $g'(9)$ .
$f$ is invertible with $f(3) = 9$ and $f'(3) = 2$ . Find $(f^{-1})'(9)$ .
$g$ is the inverse of $f$ . Which expression equals $g'(7)$ ?
1. $\dfrac{1}{f'(g(7))}$
2. $\dfrac{1}{f'(7)}$
3. $f'(g(7))$
4. $\dfrac{1}{f(7)}$
$g$ is the inverse of $f$ . Which expression equals $g'(5)$ ?
1. $\dfrac{1}{f'(g(5))}$
2. $\dfrac{1}{f'(5)}$
3. $f'(g(5))$
4. $\dfrac{1}{f(5)}$
$g$ is the inverse of $f$ . Which expression equals $g'(9)$ ?
1. $\dfrac{1}{f'(g(9))}$
2. $\dfrac{1}{f'(9)}$
3. $f'(g(9))$
4. $\dfrac{1}{f(9)}$
$g$ is the inverse of $f$ . Which expression equals $g'(4)$ ?
1. $\dfrac{1}{f'(g(4))}$
2. $\dfrac{1}{f'(4)}$
3. $f'(g(4))$
4. $\dfrac{1}{f(4)}$
$g$ is the inverse of $f$ . Which expression equals $g'(8)$ ?
1. $\dfrac{1}{f'(g(8))}$
2. $\dfrac{1}{f'(8)}$
3. $f'(g(8))$
4. $\dfrac{1}{f(8)}$
$g$ is the inverse of $f$ . Which expression equals $g'(3)$ ?
1. $\dfrac{1}{f'(g(3))}$
2. $\dfrac{1}{f'(3)}$
3. $f'(g(3))$
4. $\dfrac{1}{f(3)}$
$g$ is the inverse of $f$ . Which expression equals $g'(6)$ ?
1. $\dfrac{1}{f'(g(6))}$
2. $\dfrac{1}{f'(6)}$
3. $f'(g(6))$
4. $\dfrac{1}{f(6)}$
$g$ is the inverse of $f$ . Which expression equals $g'(2)$ ?
1. $\dfrac{1}{f'(g(2))}$
2. $\dfrac{1}{f'(2)}$
3. $f'(g(2))$
4. $\dfrac{1}{f(2)}$
A circular ripple's radius grows at 5 cm/s. How fast is its area growing when the radius is 11 cm? (cm²/s, exact)
A balloon's radius increases at 5 cm/s. With $V = \tfrac{4}{3}\pi r^3$ , find $\dfrac{dV}{dt}$ when $r = 3$ cm. (cm³/s, exact)
Air is pumped into a spherical balloon at 29 cm³/s. How fast is the radius increasing when $r = 6$ cm? (cm/s, exact)
A 15-ft ladder leans against a wall. Its foot slides away from the wall at 4 ft/s. How fast is the top moving when the foot is 9 ft from the wall? (ft/s; negative means down)
Two cars leave the same crossing at noon: one drives north at 50 mph, the other east at 51 mph. How fast is the distance between them growing? (mph, exact)
A plane flies at an altitude of 6 km, horizontally away from a radar station, at 11 km/min. How fast is its distance to the station increasing when the horizontal distance is 8 km? (km/min)
A person 6 ft tall walks away from a 13-ft street lamp at 4 ft/s. How fast is the length of their shadow growing? (ft/s, exact)
Water flows at 9 m³/min into a cone-shaped tank, vertex down, whose radius is always half the water's depth: $V = \tfrac{\pi}{12}h^3$ . How fast is the depth rising when $h = 4$ m? (m/min, exact)
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 3$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 3$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 6$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 6$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 5$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 5$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 9$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 9$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 7$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 7$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 4$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 4$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 2$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 2$ and the known rate
Put the steps for finding $\dfrac{dA}{dt}$ when $r = 8$ in order.
- Draw a picture and name the changing quantities
- Write an equation linking them: $A = \pi r^2$
- Differentiate with respect to $t$ : $A' = 2\pi r \, r'$
- Substitute $r = 8$ and the known rate

Answers

$- \frac{x}{6 y}$
$\frac{- y - 7}{x}$
$\frac{3 x^{2} + 5}{2 y}$
$- \frac{4}{3}$
$10 - 2 x$
$- \frac{4}{5}$
$\frac{1}{11}$
$\frac{1}{2}$
$\dfrac{1}{f'(g(7))}$
$110 \pi$
$180 \pi$
$\frac{29}{144 \pi}$
$-3$
$\sqrt{5101}$
$\frac{44}{5}$
$\frac{24}{7}$
$\frac{9}{4 \pi}$
1. Draw a picture and name the changing quantities 2. Write an equation linking them: $A = \pi r^2$ 3. Differentiate with respect to $t$ : $A' = 2\pi r \, r'$ 4. Substitute $r = 3$ and the known rate