Problem Sets: Derivative: Definition and Calculation (导数的定义和计算)
Published:
Last Update: 2025-10-28
Remark:
- Per the sequence of the course outline, this set does not yet cover problems related to exponential functions, inverse functions, and logarithms—these will be included in subsequent sets.
- 根据课程大纲的内容顺序安排，此部分练习暂未涵盖与指数函数、反函数及对数函数相关的题目 —— 这类题目将在后续练习中纳入。
Derivatives
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
\( y = \left(1 - \sqrt{x}\right)^{-1} \)
\( y^{2} = x^{2} + 2x \)
\( xy - y^{2} = 0 \)
\( y = \tan(x + y) \)
Suppose \( f \) is a function with the property that \( \lvert f(x)\rvert \leq x^{2} \) for all \( x \). Show that \( f(0) = 0 \).
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 \( f’(x) \).
Show that \( \dfrac{d}{dx} \left( \dfrac{\sin^{2} x}{\sec x + 1} + \dfrac{\cos^{2} x}{\sec x - 1} \right) = -\cos 2x \).
If \( f(x) = \lim_{t \to x} \dfrac{\sec t - \sec x}{t - x} \), find the value of \( f’(\pi/4) \).
Prove the product rule for three functions, that is \(\dfrac{d}{dx}(uvw) = \frac{du}{dx}vw+ u\frac{dv}{dx}w + uv\frac{dw}{dx}\).
If \( f(x) = \dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \), calculate \( f’(x) \).
Is there any special property about the derivative of an odd (or even) differentiable function? Give reasons for your answer.
Suppose \( f(x) \) is differentiable. If \( \dfrac{d}{dx}f(x^{2}) = \dfrac{d}{dx}f^{2}(x) \) when \( x = 1 \), show that either \( f’(1) = 0 \) or \( f(1) = 1 \).
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 \( f’(x) \).
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 \( f’(1) \).
Assume \( y = f(x + y) \), where \( f \) is twice differentiable and \( f’(x) \neq 1 \), find \( \dfrac{d^{2} y}{dx^{2}} \).
Suppose \( \lim_{x \to 0} \dfrac{f(x)}{1 - \cos x} = 2 \) and \( f(x) \) has a second derivative at \(x = 0 \), find \( f’‘(0) \).
Suppose that \( \lim_{x \to 0} \dfrac{\sin 6x + xf(x)}{x^{3}} = 0 \).
Find \( \lim_{x \to 0} \frac{6 + f(x)}{x^{2}} \).
Does \( f’(0) \) exist?
If \(f(x)\) is continuous, find \(f’(0)\).
If \( f(x) \) is twice-differentiable, find \( f’‘(0) \). Can the condition of be weakened and still maintain the same result?
Tangent Line, Normal Line, Differential
Assume \(f(x)\) be a differentiable function, \(f(1) = 2\), and satisfy \(\lim_{x \to 0} \dfrac{f(1) - f(1 - x)}{2x} = -1\). 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
If \( f \) and \( g \) are differentiable functions with \( f(0) = g(0) = 0 \) and \( g’(0) \neq 0 \), show that \(\lim_{x \to 0} \dfrac{f(x)}{g(x)} = \dfrac{f’(0)}{g’(0)}\).
Evaluate \( \lim_{x \to 0} \dfrac{\sin (3+x)^{2} - \sin 9}{x} \).
\(\lim_{x \to 0} \dfrac{[\sin x - \sin(\sin x)] \cdot \sin x}{x^{4}}\)
Assume that \(f(x)\) is differentiable at \(x = 1\), with \(f(1) = 2\) and \(f’(1) = 3\), find the limit \(\lim_{h \to 0} \dfrac{f(1 + h) \cdot f(1 - 3h) - 4}{h}\).
\(\lim_{x \to 3} \dfrac{\sqrt{x^{3} + 9}\sqrt[3]{2x^{2} - 17} - 6}{4 - \sqrt{x^{3} - 23}\sqrt[3]{3x^{2} - 19}}\).
\(\lim_{x \to 0} \dfrac{\sqrt{(4 + x)^{3} - 7(4 + x)} - 6}{x}\)
Find the values of the constants \( a \) and \( b \) such that \(\lim_{x \to 1} \dfrac{\sqrt[3]{ax + b} - 2}{x - 1} = \dfrac{5}{12}\).
If \( f \) is differentiable at \( a \), where \( a > 0 \), evaluate the following limit in terms of \( f’(a) \): \(\lim_{x \to a} \dfrac{f(x) - f(a)}{\sqrt{x} - \sqrt{a}}\)
Evaluate \( \lim_{x \to 0} \dfrac{\sin(a + 2x) - 2\sin(a + x) + \sin a}{x^{2}} \)
Find the constants \( a \) and \( b \) such that the following limit exists: \(\lim_{x \to 0} \dfrac{1 + a\cos 2x + b\cos 4x}{x^{4}}\). Based on the results, find the limit.
\(\lim _{x \to 0} \dfrac{x-\sin x}{3 x^{3}}\)
\(\lim _{x \to 0} \dfrac{x-\sin x}{\tan ^{3} x}\)
\(\lim _{x \to 0} \dfrac{1}{x}\left(\frac{1}{x}-\cot x\right)\)
\(n\)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?
Prove that the function \(y = f(x)\) is differentiable at \(x = x_0\) if and only if there exists a constant \(A\) such that \[\Delta y = A\Delta x + \varepsilon\Delta \] where \(\varepsilon \to 0\) as \(\Delta x \to 0\).
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 \( x = 0 \)?
If \( f(0) = 0 \), and \( f’(0) = 1 \).
Find the limit \( \lim_{x \to 0} \dfrac{f(1 - \cos x)}{\tan (x^{2})} \);
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 \( x = 0 \)?
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 \(x = 0\).