Lesson 60 — Term 6 Consolidation: Derivatives

Compute $f'(3)$ using the definition of the derivative for $f(x) = x^2$.

Compute $f'(1)$ using the definition for $f(x) = 1/x$.

Compute $f'(4)$ using the definition for $f(x) = \sqrt{x}$.

What is $f'(a)$ by the formal definition?

Let $g(x) = x^2 \cos(1/x)$ for $x \neq 0$ and $g(0) = 0$. Compute $g'(0)$ using the definition.

Compute $f'(x)$ for $f(x) = x^4 - 3x^2 + 7$.

Compute $h'(x)$ for $h(x) = e^x \sin x$.

Compute $q'(x)$ for $q(x) = \dfrac{x+1}{x-1}$.

Compute $p'(x)$ for $p(x) = x \sin x$.

Compute $r'(x)$ for $r(x) = \dfrac{\sin x}{x^2}$ and simplify.

Compute $f'(x)$ for $f(x) = x^3 e^x$ and factor the answer.

Compute $y'(2)$ for $y(x) = \dfrac{x^3}{3} - 2x + 1$.

What is the correct formula for the derivative of the product $u(x)\cdot v(x)$?

Compute $f'(x)$ for $f(x) = \dfrac{x^2}{x^2 - 1}$.

Compute $f'(x)$ for $f(x) = (2x+1)^5$.

Compute $g'(x)$ for $g(x) = \sin(x^3)$.

Compute $k'(x)$ for $k(x) = e^{x^2}$.

Compute $h'(x)$ for $h(x) = \ln(2x + 3)$.

Compute $m'(x)$ for $m(x) = \arcsin(x^2)$.

Compute $p'(x)$ for $p(x) = \cos(\sin x)$.

Compute $f'(x)$ for $f(x) = x^x$ ($x > 0$) using logarithmic differentiation.

Compute $w'(x)$ for $w(x) = \ln(\sin(3x) + 2)$.

Compute $y'$ by implicit differentiation for $x^2 + y^2 = 25$.

Compute $y'$ at $(1, 1)$ for the curve $x^3 + y^3 = 2$.

Compute $y'$ at $(0, \pi/2)$ for $\sin(xy) = y$.

Compute $y'$ for the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ and describe the tangent at $(3, 0)$.

What does it mean to "treat $y$ as an implicit function of $x$" when differentiating an equation?

Compute $y''$ implicitly for $x^2 + y^2 = r^2$.

Compute $y'$ for $x^2 y + xy^2 = 6$.

Compute $f''(x)$ for $f(x) = x^3 - 2x^2 + 2x - 1$.

Compute $f^{(4)}(x)$ for $f(x) = \sin x$.

Compute $f'(x)$ for $f(x) = \arctan x$.

Compute $g'(x)$ for $g(x) = \arctan(2x)$.

Compute $h'(x)$ for $h(x) = \ln(2x)$ and identify the most common error.

Compute $f'(x)$ for $f(x) = a^x$, with $a > 0$ and $a \neq 1$.

Compute $f^{(50)}(x)$ for $f(x) = \sin(2x)$.

Use the linearization of $f(x) = e^x$ at $a = 0$ to approximate $e^{0.1}$.

Use the linearization of $f(x) = \sqrt{x}$ at $a = 25$ to approximate $\sqrt{25.1}$.

Use the linearization of $f(x) = \ln x$ at $a = 1$ to approximate $\ln(1.05)$.

What is geometrically the linearization $L(x)$ of $f$ at $a$?

The radius of a sphere is $r = 5$ cm with a measurement error of $dr = 0.1$ cm. Use differentials to estimate the absolute and relative error in the volume $V = \tfrac{4}{3}\pi r^3$.

If the relative error in the radius of a sphere is $1\%$, what is the relative error in the volume? Justify with differentials.

The volume of a sphere is increasing at $2$ cm³/s. What is $dr/dt$ when $r = 2$ cm?

An inverted cone has radius/height ratio $r/h = 1/2$. Water flows in at $3$ m³/min. What is $dh/dt$ when $h = 2$ m?

An 8 m ladder leans against a wall. The foot of the ladder slides away from the wall at $1$ m/s. How fast is the top of the ladder sliding down the wall when the foot is $5$ m from the wall?

Two cars start from a crossing: one travels north at $40$ km/h, the other east at $30$ km/h. What is the rate of change of the distance between them when the first car has traveled $4$ km and the second $3$ km?

Show that if the radius of a circle grows at a constant rate $c$ cm/s, then the rate of change of the area is proportional to the radius $r$.

The radius of a circle is growing at $1$ cm/s. Compute the rate of change of the area when $r = 3$ cm.

Analyze the differentiability of $f(x) = |x|$ at $x = 0$.

Determine $a$ and $b$ such that $f(x) = \begin{cases} x^2, & x < 1 \\ ax + b, & x \geq 1 \end{cases}$ is of class $C^1$ at $x = 1$.