A civil engineer is analyzing the performance of a high-speed rail system. The position of a train along a straight track is given by the function $s(t) = 0.5t^3 - 2t^2 + 5t$ (where $s$ is in meters and $t$ is in seconds). The first derivative of this position function with respect to time, $s'(t) = 1.5t^2 - 4t + 5$, represents the train's instantaneous velocity. The second derivative, $s''(t) = 3t - 4$, represents its acceleration. By analyzing these derivatives, the engineer can ensure the acceleration never exceeds passenger comfort limits and verify braking distances.

In beam design, the relationship between load, shear, and moment is purely differential. If a beam is subjected to a distributed load $w(x)$, the internal shear force $V(x)$ is the negative integral of the load. More importantly, the bending moment $M(x)$ is related to the shear force by the derivative: $\frac{dM}{dx} = V(x)$. This means that wherever the shear force is zero ($V(x) = 0$), the bending moment is at a local maximum or minimum. Engineers use this derivative relationship constantly to locate the points of maximum stress in a structural member.

Find the derivative of $f(x) = 3x^2$ using the formal limit definition.

Determine if $f(x) = |x - 2|$ is differentiable at $x = 2$ using one-sided limits.

Find the derivative of the polynomial function:

Find the derivative of $y = (2x + 1)(x^2 - 3)$ using the Product Rule.

Find the derivative of the rational function using the Quotient Rule:

Find the derivative of $y = (3x^2 + 1)^5$.

Find the derivative of $y = \sqrt{x^3 - 2x + 5}$.

Find the derivative of $y = \left(\frac{2x-1}{x+3}\right)^4$. This requires both the Chain Rule and the Quotient Rule.

Find the slope of the tangent line to the circle $x^2 + y^2 = 25$ at the point $(3, 4)$.

Find $\frac{dy}{dx}$ for the equation $x^3 y + xy^3 = 10$.

Find the second derivative of $y = 4x^3 - 3x^2 + 2x - 1$.

Find the third derivative ($y'''$) of $y = \frac{1}{x}$.