Limits and Derivatives

If the slope of a curve is zero at every point, what kind of function is it?

If $f(x) = x^5$, calculate the value of $f'(1)$.

What is the derivative of the constant function $f(x) = \sqrt{2}$?

Find the derivative of $f(x) = x^2 - 1$ at $x = 3$.

Given $f(x) = \pi x^2$, find the rate of change of $f(x)$ at $x = 1$.

Find the derivative of $f(x) = \sqrt{x}$ at $x = \frac{1}{4}$.

Which of the following is the result of $\lim_{h \to 0} \frac{(x+h)^3 - x^3}{h}$?

If $f(x) = \sqrt{x} + \frac{1}{\sqrt{x}}$, find $f'(1)$ using the definition of derivatives.

Find the slope of the normal to the curve $y = x^2$ at $x=1/2$.

Let $f(x) = |x-1| + |x+1|$. The derivative $f'(x)$ at $x=0$ is:

Find $\lim_{x \to 0} (1 + x + x^2)$.

Evaluate $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$.

What is $\lim_{x \to -1} \frac{x^3 + 1}{x + 1}$?

Evaluate the limit: $\lim_{x \to 1} \frac{x^2 - 3x + 2}{x - 1}$.

Evaluate the limit: $\lim_{x \to 0} \frac{1 - \cos 6x}{x \tan 3x}$

Calculate $\lim_{x \to 0} \frac{\sin x (1 - \cos x)}{x^3}$.

Evaluate $\lim_{x \to 0} \frac{1 - \cos m x}{x^2}$.

If $f(x) = \frac{|x|}{x}$, find $\lim_{x \to 0} f(x)$

Evaluate $\lim_{x \to 0} \frac{\tan 3x - 2x}{3x - \sin^2 x}$

Find the limit: $\lim_{x \to \infty} \frac{\sin x}{x}$

What is the value of $\lim_{x \to 0} \frac{\tan x - \sin x}{x}$?

Find $\lim_{x \to 0} \frac{x^2}{\sin x \tan x}$.

Evaluate $\lim_{x \to 0} \frac{1 - \cos 2x}{x}$.

Calculate $\lim_{x \to \pi} \frac{\sin x}{\pi - x}$.

Compute $\lim_{x \to 0} \frac{1 - \cos x \sqrt{\cos 2x}}{x^2}$.

Find $\lim_{x \to \pi/2} \frac{1 - \sin x}{(\pi/2 - x)^2}$.

Evaluate $\lim_{x \to 0} \frac{\sec x - 1}{x^2}$.

In a physics problem involving diffraction, the intensity $I$ is related to $\lim_{x \to 0} \frac{\sin(x) - x + \frac{x^3}{6}}{x^5}$. Find the value of this limit.

Determine the limit $\lim_{x \to 0} \frac{\sqrt{1 + \sin(x)} - \sqrt{1 - \sin(x)}}{x}$ to find the linearization constant of a sensor reading.

A satellite trajectory calculation requires the value of $\lim_{x \to 0} \frac{x^2 \cot(x)}{\sin(x)}$. What is the resulting value?

Find the derivative of $f(x) = (x-1)(x-2)$.

Calculate $\frac{d}{dx} (x \cos x)$.

Determine the derivative of $f(x) = x^2 - 2\sin x$.

If $f(x) = \frac{\cos x}{x}$, find $f'(x)$.

If $f(x) = \frac{x}{x+1}$, find the value of $f'(0)$.

Differentiate $f(x) = a\sin x + b\cos x + c$ where $a, b, c$ are constants.

What is the derivative of $f(x) = \frac{x^3}{\sin x}$?

Find the derivative of $f(x) = \frac{\sin(x+\alpha)}{\sin(x-\alpha)}$.

What is the derivative of $f(x) = \frac{x \tan x}{\sec x + \tan x}$?

Find the derivative of $f(x) = \frac{\sin x}{x + \cos x}$.

The graph of the function $f(x) = x^2 + 2$ is shown below. According to the algebra of derivatives, if $h(x) = f(x) + g(x)$ where $g(x) = x^3$, find the value of the derivative $h'(x)$ at $x = 1$.

Find $\frac{d}{dx}(x^2 - x + 1)$.

Find the derivative of $f(x) = 100$.

What is the derivative of $f(x) = x - \cos x$?

Find the derivative of $f(x) = \frac{1}{x+1} + \frac{1}{x-1}$.

Using the product rule, find $\frac{d}{dx}(x^3 \cos x)$.

Find the derivative of $f(x) = 3x^2 \cdot e^x$.

Calculate the derivative of the function $f(x) = \frac{x^2 - 4}{x^2 + 4}$ at $x = 2$.

The pressure $P$ and volume $V$ of a gas are related such that $P = \frac{k}{V^2}$. If $k=100$, find the rate of change of pressure with respect to volume $\frac{dP}{dV}$ when $V = 5$.

Let $h(x) = (x^2 + x + 1)(x - 1)$. Find $h'(x)$ at $x = 0$ using the product rule.

If $\lim_{x \to a} f(x) = 2$, find the value of $\lim_{x \to a} [f(x)]^3$.

If $\lim_{x \to a} f(x) = 8$, find the value of $\lim_{x \to a} \frac{f(x)}{4}$.

Evaluate the limit: $\lim_{x \to 1} (4x^3)$.

Evaluate the limit: $\lim_{x \to 0} \frac{\tan 3x}{3x}$.

Evaluate $\lim_{x \to 1} \frac{x^m - 1}{x^n - 1}$ where $m, n$ are positive integers.

Calculate the limit: $\lim_{x \to 0} \frac{(1+x)^6 - 1}{(1+x)^2 - 1}$

Evaluate the limit: $\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{2x}$

Calculate the limit: $\lim_{x \to 0} \frac{8}{x^8} \left( 1 - \cos \frac{x^2}{2} - \cos \frac{x^2}{4} + \cos \frac{x^2}{2} \cos \frac{x^2}{4} \right)$.

Evaluate the limit: $\lim_{x \to 0} \frac{\sqrt{1+x+x^2} - \sqrt{x+1}}{x^2}$.

Find $\lim_{x \to \pi/4} \frac{\sec^2 x - 2}{\tan x - 1}$.

The graph of a quadratic polynomial function $f(x) = x^2 - 1$ is shown below. Find the value of $\lim_{x \to 2} f(x)$ using the properties of limits for polynomials.

Find $\lim_{x \to 2} (x^3 - 2x^2 + 4x - 8)$.

Calculate $\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$.

Find $\lim_{x \to 1} [ (x-1)^2 + 5 ]$.

Compute $\lim_{x \to 1} \frac{x^7 - 1}{x^4 - 1}$.

Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x^2 + 3x - 10}$.

Find the value of $\lim_{x \to 0} \frac{(1+x)^n - 1}{x}$.

A civil engineer is designing a curved ramp for a highway interchange. The profile of the ramp is modeled by the rational function $f(x) = \frac{x^n - a^n}{x - a}$, where $a = 2$ and $x$ represents the horizontal distance in decameters. To ensure the safety of the transition at the junction where $x$ approaches $a$, the engineer needs to determine the theoretical height limit. If the calculated limit at $x = 2$ is exactly $80$ units, what is the integer value of the power $n$ used in the design?

An experimental data set follows the curve $y = \frac{1 - x^m}{1 - x^n}$ near $x = 1$. If $m = 6$ and $n = 2$, calculate the value of $\lim_{x \to 1} y$.

Given that $\lim_{x \to 1} \frac{ax^2 + b}{x - 1} = 4$, where $a$ and $b$ are constants, find the values of $a$ and $b$.

Differentiate $f(x) = \sqrt{x}$ with respect to $x$.

The derivative of $f(x) = 2\sin x$ is:

Find the derivative of $f(x) = x^n$ at $x=1$.

What is $\frac{d}{dx}(\cos x)$ at $x = \frac{\pi}{2}$?

The graph of the function $f(x) = x^2 - 4x + 3$ is shown in the coordinate plane. Find the value of the derivative $f'(x)$ at the point where the curve intersects the y-axis.

Differentiate $f(x) = \sin^2 x$ by using product rule on $(\sin x)(\sin x)$.

Compute the derivative of $f(x) = \frac{2}{x+1} - \frac{3}{x}$.

A projectile's path is given by $f(x) = x \tan(\theta) - \frac{gx^2}{2u^2 \cos^2(\theta)}$. If $\theta = 45^\circ$ and $\frac{g}{2u^2 \cos^2(\theta)} = 0.01$, find the slope of the path at $x = 10$.

The position of a oscillating spring is described by $y(t) = \frac{\sin(t)}{t+1}$ for $t \ge 0$. Find the derivative $y'(t)$ at $t = 0$.

A water tank is in the shape of a cone. The volume of water $V$ in terms of the depth $h$ is given by $V(h) = \frac{\pi}{12} h^3$. If the depth of water is increasing, find the rate of change of volume with respect to the depth when $h = 2$ units.