Exercises: Exercises on derivatives

Calculate the first and second derivatives of the polynomial function $f(x) = 3x^4 - 5x^3 + 2x - 7$.

Find the derivative of the function $f(x) = (2x^2 + 3) e^x$.

Determine the derivative of the rational function $f(x) = \frac{3x - 1}{x^2 + 4}$.

Calculate the derivative of the function $f(x) = (5x^3 - 2x)^4$ using the chain rule.

Find the derivative of the trigonometric function $f(x) = \cos(x^3 - 4x)$.

Calculate the derivative of the function $f(x) = \ln(7x^2 + 5) + 3^{2x}$.

Find the derivative of the irrational function $f(x) = \sqrt{x^4 - 3x^2 + 2}$.

Calculate the derivative of the function $f(x) = \frac{2}{\sqrt[3]{x}}$. Express the result as a fractional power or a root.

The electric charge passing through a circuit as a function of time is given by the expression $q(t) = 5t e^{-0.2t}$ coulombs. Calculate the electric current $I(t) = q'(t)$ and determine the exact time at which the current is zero.

Calculate the derivative of the function $f(x) = \frac{\sqrt{x^2-1}}{x+2}$ and simplify the result as much as possible, expressing it with a single denominator.

The population of a species of marine alga, in millions of individuals, follows the function $P(t) = \ln(t^2 + 4t + 1) + 5$, where $t$ is the time in weeks. Determine the instantaneous growth rate of the population in the third week ($t=3$).

The total cost in pesos of producing $x$ custom sweatshirts for the end-of-year trip is given by $C(x) = \frac{3\,000x}{\sqrt{x^2+100}} + 4\,000$. Calculate the marginal cost function $C'(x)$.

Calculate the derivative of the composite function $f(x) = \sin^3(e^{2x} + 1)$, taking care with the successive application of the chain rule.

Given the function $f(x) = \ln\left(\cos(3x^2)\right)$, calculate its derivative and evaluate it at the point $x = \sqrt{\frac{\pi}{12}}$.

The current in an alternating current circuit fluctuates according to the function $I(t) = \frac{\sin(3t)}{t^2 + 2}$, where $t$ is the time in seconds. Calculate the derivative $I'(t)$ applying the quotient rule and the chain rule.

Show, using the laws of logarithms and the rules of differentiation, that the derivative of the function $f(x) = \ln \sqrt{\frac{1+\sin x}{1-\sin x}}$ is $f'(x) = \frac{1}{\cos x}$.

Calculate the nth derivative of the function $f(x) = \frac{1}{2x-1}$. Express the result $f^{(n)}(x)$ as a general formula in terms of $n$.

Given two differentiable functions $g(x)$ and $h(x)$, a new function $f(x) = g(h(x^2))$ is defined. Given that $h(1) = 2$, $h'(1) = 3$, $g(2) = 5$, and $g'(2) = -4$, calculate the exact value of $f'(1)$, showing the steps of the chain rule in detail.

Find the values of the parameters $a$ and $b$ (with $b > 0$) such that the function $f(x) = a e^{bx}$ simultaneously satisfies $f(0) = 3$ and that its second derivative evaluated at the origin is $f''(0) = 12$.

Calculate the derivative of the function $f(x) = \arcsin\left(\frac{1-x^2}{1+x^2}\right)$ for $x > 0$ and simplify the resulting expression as far as possible.

The cooling law of a ceramic material in an industrial furnace follows the function $T(t) = 20 + 800 e^{-0.05t}$, where $T$ is the temperature in degrees Celsius and $t$ is the time in minutes. Calculate the rate of change of temperature with respect to time, $T'(t)$, and determine how much time must elapse for the temperature to decrease at an exact rate of $10^\circ \text{C/min}$.

Calculate the derivative of the following function, using the product rule and the chain rule: $f(x) = e^{x\sin(x)} \times \ln(x^2 + 4)$.

Show, by calculating the derivative and studying its sign, that the function $f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ is strictly increasing throughout its domain.