In urban planning and water resources engineering, estimating future population is critical for designing water treatment plants and distribution networks. Often, population growth is modeled exponentially as $P(t) = P_0 e^{kt}$, where $P_0$ is the initial population and $k$ is the growth rate constant. The derivative, $\frac{dP}{dt} = k P_0 e^{kt} = k P(t)$, represents the instantaneous rate of population change. This simple differential relationship—that the rate of growth is directly proportional to the current population—forms the basis for capacity planning over a 20- or 50-year design horizon.

Civil engineers frequently design suspension bridges and power transmission lines. When a uniform cable hangs under its own weight between two supports, it does not form a parabola; it forms a curve called a catenary, which is modeled by the hyperbolic cosine function: $y(x) = a \cosh(\frac{x}{a})$. The derivative of this shape, $y'(x) = \sinh(\frac{x}{a})$, gives the slope of the cable at any point $x$. This slope is critical for determining the tension vector in the cable and designing the anchorage systems that secure it to the ground.

Find the derivative of a simple composite function: $y = \sin(3x^2)$.

Find the derivative involving a product of trigonometric functions: $f(x) = x^2 \tan x$.

Find the second derivative of $y = \cos(2x)$.

Differentiate the function: $y = \arctan(3x^2)$.

Find the derivative of $y = \arcsin(e^x)$.

Find the rate of change of the population function $P(t) = 100e^{0.05t}$.

Differentiate a logarithmic function with an inner polynomial: $y = \ln(x^2 + 1)$.

Find the derivative of a general exponential function: $y = 5^{x^2}$.

Use logarithmic differentiation to find the derivative of $y = x^{\sin x}$.

Use logarithmic differentiation to simplify finding the derivative of $y = \frac{(x+1)^4 \sqrt{x-2}}{(x^2+3)^5}$.

Find the derivative of the hyperbolic function: $y = \sinh(4x^3)$.

Find the derivative of the inverse hyperbolic function: $y = \text{arcsinh}(3x)$.