Practice Questions

Identify the derivative of the natural exponential function e^x with respect to x.

Propose a modification to the function $f(x) = \frac{x^2 - 9}{x-3}$ to make it continuous at $x=3$. Justify your proposal.

Solve for $\frac{dy}{dx}$ if $y = \sin^{-1}(x) + \cos^{-1}(x)$.

Justify whether the converse of the theorem 'Every differentiable function is continuous' is true. If it is not true, provide a counterexample.

Calculate the second derivative of $y = \log(\sin x)$.

Define the continuity of a real function f at a point c in its domain.

Explain the geometric meaning of a function being not continuous at a point x=c.

Formulate a function that is continuous for all real numbers but fails to be differentiable at the specific point $x=a$.

State the product rule (also known as the Leibnitz rule) for the differentiation of two functions u(x) and v(x).

Recall the derivatives of the following inverse trigonometric functions with respect to x: (a) \tan^{-1} x (b) \sin^{-1} x

State the derivative of \cos^{-1} x with respect to x and specify its domain.

Analyze the function defined by $f(x) = \begin{cases} \frac{x^2 - 9}{x-3}, & \text{if } x \neq 3 \ k, & \text{if } x = 3 \end{cases}$ and calculate the value of $k$ that makes $f(x)$ continuous at $x=3$.

Summarize the technique of logarithmic differentiation. Explain the type of functions for which this method is particularly useful and identify the necessary conditions for its application.

Critique the following statement and justify your conclusion: 'If two functions $f(x)$ and $g(x)$ are both discontinuous at a point $x=c$, then their sum, $(f+g)(x)$, must also be discontinuous at $x=c$.'

Evaluate the differentiability of the function $f(x) = |x| + |x-1|$ at $x=0$ and $x=1$ by justifying the geometric interpretation of the graph.

Prove that the function $f(x) = |\cos x|$ is continuous for all $x \in \mathbb{R}$ but is not differentiable at $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer.

Derive a general formula for the derivative of the product of three differentiable functions, $h(x) = f(x)g(x)k(x)$, by applying the product rule twice. Justify each step.

Evaluate the values of $p$ and $q$ for which the function $f(x)$ is differentiable at $x=1$. $f(x) = \begin{cases} x^2 + 3x + p, & x \leq 1 \ qx + 2, & x > 1 \end{cases}$

Design a function that is continuous for all real numbers but is not differentiable at exactly two points, $x=-2$ and $x=2$. Provide the function and prove that it satisfies these conditions.

State the Chain Rule for differentiating a composite function. If f is a composite of two functions u and v such that f = v \circ u, what is the derivative of f?

Calculate the derivative of $f(x) = e^{\cos(x^2)}$ with respect to $x$.

Examine if the function $f(x) = |x-2| + |x+1|$ is continuous at $x=2$.

Calculate $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$.

State the theorem regarding the algebra of continuous functions. If f and g are two real functions continuous at a real number c, list the continuity of their sum, difference, product, and quotient.

Recall the definition of the derivative of a function f at a point c using the limit definition.

Explain why every polynomial function is continuous for all real numbers.

Describe in detail the three conditions that must be satisfied for a function f to be continuous at a point x = c. Explain what each condition means.

Calculate $\frac{dy}{dx}$ for the function $y = \tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$.

If $y = A e^{2x} + B e^{-x}$, solve the differential equation $\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0$.

If $x = a(t - \sin t)$ and $y = a(1 - \cos t)$, calculate $\frac{dy}{dx}$.

Calculate the derivative of $y = x^{\cos x}$ with respect to $x$.

What is an implicit function? Provide one example.

Describe the concept of a second-order derivative. List at least three different notations used to represent it.

Demonstrate that the function $f(x) = |x-3|$ is not differentiable at $x=3$.

If $f(x)$ is a differentiable function with a differentiable inverse $g(x)$ (so that $f(g(x)) = x$), derive a formula for the second derivative of the inverse function, $g''(x)$, in terms of the derivatives of the original function, $f'$ and $f''$. Justify each step of the derivation.

Critique the following attempt to differentiate $y = x^{\sin x}$. A student takes the logarithm of both sides and proceeds as follows: $\log y = \log(x^{\sin x}) = (\sin x)(\log x)$. Then, the student incorrectly assumes that the derivative of a product of functions is the product of their derivatives, writing $\frac{1}{y}\frac{dy}{dx} = (\cos x)(\frac{1}{x})$. Identify the conceptual error and provide the correct, complete solution.

Justify the relationship that must exist between the constants $a$ and $b$ for the function $f(x)$ defined below to be continuous at $x=0$. $f(x) = \begin{cases} \frac{1 - \cos(ax)}{x \sin x}, & x \neq 0 \ b, & x=0 \end{cases}$

Explain the relationship between differentiability and continuity at a point. State the relevant theorem and provide a well-known example to show that the converse of the theorem is not true.

Calculate the derivative of $y = (\log x)^x + x^{\sin x}$ with respect to $x$.

If $y = (\tan^{-1} x)^2$, demonstrate that $(x^2+1)^2 y_2 + 2x(x^2+1)y_1 - 2 = 0$, where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$.

If $x^y = e^{x-y}$, calculate $\frac{dy}{dx}$.

Given the function $y = (x + \sqrt{x^2 + 1})^m$, prove the relation $(x^2+1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - m^2y = 0$.

Formulate the second derivative of $y$ with respect to $x$, given the parametric equations $x = a(\theta - \sin \theta)$ and $y = a(1 - \cos \theta)$.

If $x = e^{\sin(2t)}$ and $y = e^{\cos(2t)}$, prove that $\frac{dy}{dx} = -\frac{y \log x}{x \log y}$.