Problem Sets: Differentiation (微分)
Published:
Last Update: 2026-10-19
Remark:
In accordance with the course sequence (late-transcendentals), this problem set does not yet include problems involving exponential functions, inverse functions, or logarithms. These topics will be covered in subsequent problem sets.
The Additional and Advanced Exercises are somewhat more involved than the standard exercises. They may require multi-step reasoning, conceptual analysis, modest proof components, or a combination of several ideas.
Derivatives
Differentiation Rules and Implicit Differentiation
Find $\dfrac{dy}{dx}$ if
$y = \dfrac{\tan 3x}{(x + 7)^{4}}$
$x + \tan(xy) = 10$
$y = \sqrt{\dfrac{x - 1}{x - 3}}$
$y = \sqrt{x \sin(x\sqrt{1 - x})}$
$y = \sin(\cos(\tan x))$
$y = \dfrac{3x^{7/5} + x^{3}\sqrt[3]{4x^{7} + 11 x^{4/7} - 23x + 9}}{6x^{2} + 4}$
$f(x) = \begin{cases} x^{2} \sin\dfrac{1}{x}, & x > 0, \ 0, & x = 0, \ \dfrac{1 - \cos x^{2}}{x}, & x < 0. \end{cases} $
Find $\dfrac{d^{2}y}{dx^{2}}$ if
$y = \left(1 - \sqrt{x}\right)^{-1}$
$y^{2} = x^{2} + 2x$
$xy - y^{2} = 0 $
$y = \tan(x + y)$
Show that $\dfrac{d}{dx} \left( \dfrac{\sin^{2} x}{\sec x - 1} + \dfrac{\cos^{2} x}{\sec x + 1} \right) = -\cos 2x$.
If $f(x) = \dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}$, calculate $f’(x)$.
Prove the product rule for three functions, that is $\dfrac{d}{dx}(uvw) = \dfrac{du}{dx}vw+ u\dfrac{dv}{dx}w + uv\dfrac{dw}{dx}$.
If $f(t) = \left( \tan\frac{\pi t}{4} - 1 \right)\left( \tan\frac{\pi t^{2}}{4} - 2 \right)\cdots\left( \tan\frac{\pi t^{100}}{4} - 100 \right)$, find $f’(1)$.
Is there any special property about the derivative of an odd (or even) differentiable function? Give reasons for your answer.
Assume $y = f(x + y)$, where $f$ is twice differentiable and $f’(x+y) \neq 1$, find $\dfrac{d^{2} y}{dx^{2}}$.
The Limit Defintion of Derivatives, Differentiate
Let $f(x) = \begin{cases} \cos \dfrac{\pi x}{2}, & \lvert x \rvert \leq 1, \\ \lvert x - 1\rvert, & \lvert x\rvert > 1.\end{cases}$. Find \( f’(x) \).
Suppose $f$ is a function with the property that $\lvert f(x)\rvert \leq x^{2}$ for all $x$. Show that $f$ is differentiable at $x=0$, and find $f’(0)$
If $\displaystyle f(x) = \lim_{t \to x} \dfrac{\sec t - \sec x}{t - x}$, find the value of $f’(\pi/4)$.
Suppose $f(x)$ is differentiable. If $\dfrac{d}{dx}f(x^{2}) = \dfrac{d}{dx}\left[f(x)\right]^2$ when $x = 1 $, show that either $f’(1) = 0$ or $f(1) = 1$.
Suppose $\displaystyle \lim_{x \to 0} \dfrac{f(x)}{1 - \cos x} = 2$ and $f(x)$ has a second derivative at $x = 0$, find $f’‘(0)$.
Tangent Line, Normal Line, Differential
Assume $f(x)$ be a differentiable function, $f(1) = 2$, and satisfy $\displaystyle \lim_{x \to 0} \dfrac{f(1) - f(1 - x)}{2x} = -1$. Find the equations of the tangent line and normal line to the curve $y = f(x)$ at the point $(1, 2)$.
Find $\Delta y$ and differential $dy$ of the functions $y = x^{2}$ and $y = \dfrac{x}{1 + x^{2}}$ at $x = 0$ when $\Delta x = 0.1$ respectively.
Find the tangent line to the curve $x^{1/2} + y^{1/2} = 1$ at any point. Find the intersection points $(x_0, 0)$ and $(0, y_0)$ of the tangent line with $x$-and $y$-axis, respectively, then show that $x_0 + y_0 = 1$.
Find all values of $c$ such that the parabolas $y = 4x^{2}$ and $x = c + 2y^{2}$ intersect each other at right angles.
How many lines are tangent to both of the circles $x^{2} + y^{2} = 4$ and $x^{2} + (y - 3)^{2} = 1$?
Find the tangent line of the curve $2x^{2} + x - y^{2} = 2$ at $x = 1$.
Find Limits
Evaluate $\displaystyle \lim_{x \to 0} \dfrac{\sin (3+x)^{2} - \sin 9}{x}$.
$\displaystyle \lim_{x \to 0} \dfrac{\left(\sin x - \sin(\sin x)\right) \cdot \sin x}{x^{4}}$
Assume that $f(x)$ is differentiable at $x = 1$, with $f(1) = 2$ and $f’(1) = 3$, find the limit $\displaystyle \lim_{h \to 0} \dfrac{f(1 + h) \cdot f(1 - 3h) - 4}{h}$.
$\displaystyle \lim_{x \to 0} \dfrac{\sqrt{(4 + x)^{3} - 7(4 + x)} - 6}{x}$
Find the values of the constants $a$ and $b$ such that $\displaystyle \lim_{x \to 1} \dfrac{\sqrt[3]{ax + b} - 2}{x - 1} = \dfrac{5}{12}$.
If $f$ is differentiable at $a$, where $a > 0$, evaluate the following limit in terms of $f’(a)$: $\displaystyle \lim_{x \to a} \dfrac{f(x) - f(a)}{\sqrt{x} - \sqrt{a}}$
Evaluate $\displaystyle \lim_{x \to 0} \dfrac{\sin(a + 2x) - 2\sin(a + x) + \sin a}{x^{2}}$
Find the constants $a$ and $b$ such that the following limit exists: $\displaystyle\lim_{x \to 0} \dfrac{1 + a\cos 2x + b\cos 4x}{x^{4}}$. Based on the results, find the limit.
$\displaystyle \lim _{x \to 0} \dfrac{x-\sin x}{3 x^{3}}$
$\displaystyle \lim _{x \to 0} \dfrac{1}{x}\left(\frac{1}{x}-\cot x\right)$
$\displaystyle\lim_{x \to \infty} x^{2} \left(2 - x \sin \dfrac{1}{x} - \cos \dfrac{1}{x}\right)$
$\displaystyle \lim_{x\to 0}\frac{x\tan x-\sin^{2} x}{x^{4}}$
$\displaystyle \lim_{x\to 0}\left(\frac{1}{\sin^2 x}-\frac{\cos^2 x}{x^2}\right)$
$n$th Derivatives
If $f(x) = \dfrac{1}{x^{2} + 5x + 6}$, find $f^{(n)}(x)$.
If $f(x) = \dfrac{1}{1- x - x^{2}}$, find $f^{(n)}(x)$.
If $f(x) = \dfrac{x^{46} + x^{45} +2}{1+x}$, calculate $f^{(n)}(3)$.
Find the first four derivatives of $y = \tan x$ and $y = \sec x$.
Find the $n$th derivatives of $y = \sin x$ and $y = \cos x$.
If $f(x) = \sin^{6} x + \cos^{6} x$, find $f^{(n)}(x)$.
Differentiability
Determine the value of $a, b$ for which the function $f(x) = \begin{cases} \sin x, & x < \pi, \\ ax + b, & x \geq \pi. \end{cases}$ is differentiable at $x = \pi$.
Find values for the constants $a$, $b$, and $c$ that will make $f(x) = \cos x$ and $g(x) = a + bx + cx^2$ satisfy $f(0) = g(0)$, $f’(0) = g’(0)$, and $f’‘(0) = g’‘(0)$. For the determined values of $a$, $b$, and $c$ what happens for the third and fourth derivatives of $f$ and $g$?
Find values for $b$ and $c$ that will make $f(x) = \sin(x + a)$ and $g(x) = b\sin x + c\cos x$ satisfy $f(0) = g(0)$ and $f’(0) = g’(0)$. For the determined values of $a$, $b$, and $c$ what happens for the higher-order derivatives of $f$ and $g$?
Suppose $f(x) = \begin{cases} \dfrac{a(\cos x - 1)}{\sin^{2} x}, & x < 0, \\ ax^{2} + bx + 1, & x \geq 0 \end{cases}$ is differentiable. Find the values of $a$ and $b$.
Let $f(x) = \begin{cases} 3 + x^{2}, & x \leq 0, \\ \dfrac{\sin 3x}{x}, & x > 0. \end{cases}$. Does $f’(0)$ exist?
Suppose that $\varphi(x)$ is continuous at $x = a$, study the differentiability of $f(x) = \lvert x - a\rvert \varphi(x)$ at $x = a$.
If $f(x) = \begin{cases} g(x)\cos\dfrac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, and $g(0) = g’(0) = 0$. Is the function $f(x)$ differentiable at $x = 0$?
Determine the value of $a, b$ if $f(x)$ is differentiable: $f(x) = \begin{cases} 2\sin x + a, & x < 0, \\ x^{3} + bx + 3, & x \geq 0.\end{cases}$.
Let $f(x) = \begin{cases} x^{2} \sin \dfrac{1}{x}, & x \neq 0, \\ 0, & x = 0. \end{cases}$
Is $f(x)$ continuous at $x = 0$?
Does $f’(0)$ exist?
Is $f’(x)$ continuous at $x = 0$?
Let the function $f(x)$ be continuous on $[-1,1]$, and $\displaystyle \lim_{x\to 0}\frac{f(x)}{x}=2025$. Is $f(x)$ differentiable at the origin? If so, find $f’(0)$.
If $f(0) = 0$, and $f’(0) = 1$.
Find the limit $\displaystyle \lim_{x \to 0} \dfrac{f(1 - \cos x)}{\tan (x^{2})}$;
Given function $g(x) = \begin{cases} x^{2} \sin \dfrac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, is the function $(f \circ g)(x)$ differentiable at $x = 0$?
Additional and Advanced Exercises
$\displaystyle \lim_{x \to 3} \dfrac{\sqrt{x^{3} + 9}\sqrt[3]{2x^{2} - 17} - 6}{4 - \sqrt{x^{3} - 23}\sqrt[3]{3x^{2} - 19}}$.
$\displaystyle \lim _{x \to 0} \dfrac{\sin x-\sin(\sin x)}{x^{3}}$
Let $f(x)$ defines on $(0, \infty)$, and for any $x > 0, y > 0$, it satisfies $f(xy) = f(x) + f(y)$. Moreover, $f’(1)$ exists and equals $a$. Find $f’(x)$.
Suppose $f(x)$ has a continuous first-order derivative at $x=1$, and $f’(1)=-2$. Evaluate the limit \(\lim_{x\to 0^+}\frac{d}{dx}f(\cos\sqrt{x})\)
Suppose that $\displaystyle \lim_{x \to 0} \dfrac{\sin 6x + xf(x)}{x^{3}} = 0$.
Find $\displaystyle \lim_{x \to 0} \frac{6 + f(x)}{x^{2}}$.
Does $f’(0)$ exist?
If $f(x)$ is continuous, find $f’(0)$.
If $f(x)$ is twice-differentiable, find $f’‘(0)$. Can the condition be weakened while preserving the same conclusion?
Let $f(x) = \begin{cases} x^{\lambda} \cos\dfrac{1}{x}, & x \neq 0, \\ 0, & x = 0, \end{cases}$. If $f’$ is continuous at $x = 0$, find $\lambda$.
Let $f(x) = \begin{cases} x^{\alpha} \cos\frac{1}{x^{\beta}}, & x > 0, \\ 0, & x \leq 0 \end{cases}$ ($\alpha > 0, \beta > 0$).
What conditions should $\alpha$ and $\beta$ satisfy to make $f(x)$ differentiable at $x = 0$, and find $f’(x)$;
What conditions should $\alpha$ and $\beta$ satisfy to make $f’(x)$ continuous at $x = 0$.
If $f$ and $g$ are differentiable functions with $f(0) = g(0) = 0$ and $g’(0) \neq 0$, prove the following result using the definition of the derivatives \(\displaystyle \lim_{x \to 0} \dfrac{f(x)}{g(x)} = \dfrac{f'(0)}{g'(0)}\) Note: This is a single-point version of the 0/0 limit result. It only assumes differentiability of $f,g$ at $x=0$, and does not require differentiability on a punctured neighbourhood of 0. This is not the L’Hôpital’s rule, which requires stronger hypotheses.
Prove that the function $y = f(x)$ is differentiable at $x = x_0$ if and only if there exists a constant $A$ such that \(\Delta y = A\Delta x + \varepsilon\Delta x\) where $\varepsilon \to 0$ as $\Delta x \to 0$.