Derivative exercises

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 $f(x) = \frac{3x - 1}{x^2 + 4}$.

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.

Given the rational function $f(x) = \frac{8x}{x^2 + 4}$:

1. a) Notice that $f(0) = 0$ and that for a nearby value to the right, such as $x = 0.5$, the function takes the value $f(0,5) \approx 0.94 > 0$. From this behavior, estimate with reasoning the sign that the derivative $f'(0)$ should have and state whether you expect the function to be increasing or decreasing in that neighborhood.
2. b) Apply the quotient rule to find the algebraic expression for $f'(x)$. Simplify the numerator as much as possible by factoring out a numerical common factor and factoring the resulting difference of squares.
3. c) Evaluate $f'(0)$ in the expression obtained and check whether it matches your estimate from section a).
4. d) Solve the equation $f'(x) = 0$ using the factored form of the numerator and interpret what geometric meaning the corresponding points have on the graph of $f$.

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.

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

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

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

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

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

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.

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.

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

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.

Consider the function with real parameter $a$: $f(x) = (x^2 + a) \times e^{2x}$.

1. a) Looking at the function "from the outside in", identify the main outer operation and apply the corresponding rule combined with the chain rule to calculate $f'(x)$. Express the result by factoring out the exponential $e^{2x}$ so that it is in the form $f'(x) = (2x^2 + 2x + 2a) \times e^{2x}$.
2. b) Knowing that the tangent line to the graph of $f$ at the point with abscissa $x = 1$ is horizontal (that is, its slope is zero), set up an equation involving $f'(1)$ and deduce with reasoning the value of the parameter $a$.
3. c) Using the value of $a$ found, completely factor the polynomial that multiplies $e^{2x}$ in $f'(x)$ and find all the points where the slope of the tangent line to the curve is zero.

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 euros of producing $x$ custom sweatshirts for the end-of-year trip is given by $C(x) = \frac{150x}{\sqrt{x^2+100}} + 200$. Calculate the marginal cost function $C'(x)$.

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 $f''(0) = 12$.

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: $f(x) = e^{x\sin(x)} \times \ln(x^2 + 4)$.

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.

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

In a 5th Year exam, Aoife has to calculate the derivative function of $f(x) = \frac{x^3}{x^2 + 4}$. Her written working is as follows:

Next, Aoife crosses out the $x^4$ terms from the numerator and denominator and concludes that the final result is $f'(x) = \frac{12x^2}{16} = \frac{3x^2}{4}$.

1. a) Identify and explain in detail the two serious algebraic errors Aoife made: one when expanding the denominator and another when simplifying the fraction.
2. b) Correct the exercise by computing the derivative properly, factoring out a common factor in the numerator, and leaving the expression as an irreducible fraction with the denominator factored.

According to records from Met Éireann, during a heatwave day in Cork, the temperature $T$ (in degrees Celsius) recorded between 8:00 and 22:00 can be approximated by the model:

where $t$ is the hour of the day ($8 \le t \le 22$).

1. a) Calculate the derivative function $T'(t)$ by applying the quotient rule and the chain rule. Simplify the numerator by factoring the resulting expression until you obtain $T'(t)$ as a single irreducible fraction with the numerator fully factored.
2. b) The derivative $T'(t)$ measures the instantaneous rate of change of temperature in °C/h. Calculate $T'(11)$ and interpret the result in the real-world context: is the temperature increasing or decreasing at 11:00, and at what exact rate?
3. c) Determine at what exact time of day the maximum temperature is reached by solving the equation $T'(t) = 0$, and calculate the value of this record temperature.

To differentiate the function $f(x) = \frac{5x^4 - 10x^2 + 3}{5}$, two students propose different approaches:

• Seán wants to apply the quotient rule, treating the numerator as $u(x) = 5x^4 - 10x^2 + 3$ and the denominator as the constant function $v(x) = 5$.

• Aoife proposes rewriting the function beforehand by splitting the common denominator into a sum of monomials: $f(x) = x^4 - 2x^2 + \frac{3}{5}$, and differentiating term by term.

1. a) Differentiate the function following Seán's method and simplify the result.
2. b) Differentiate the function following Aoife's method.
3. c) Compare both methods: which one requires fewer algebraic operations and has a lower risk of error? Justify what general rule regarding multiplicative constants makes it unnecessary to use the quotient rule when the denominator is a fixed number.

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 $n$.

We want to calculate the derivative function of $f(x) = \ln\left(\frac{x^2}{\sqrt{2x + 1}}\right)$, defined for $x > 0$.

1. a) Aoife tries to differentiate by directly applying the chain rule to the entire expression: first the derivative of the natural logarithm $\frac{u'}{u}$, and then the quotient rule with a square root in the denominator to find $u'$. Describe what difficulties and algebraic complexity this approach presents.
2. b) Cian proposes an alternative strategy: before differentiating, rewrite $f(x)$ by applying the properties of logarithms (logarithm of a quotient, of a power, and of a root) to decompose the function into a sum and difference of simple logarithms. Apply Cian's strategy to transform $f(x)$.
3. c) Differentiate term by term the expression simplified by Cian, bring the fractions to a single common denominator, and observe how direct the process is.
4. d) From the derivative obtained in section c), analyze whether there is any point with a horizontal tangent ($f'(x) = 0$) within the domain of definition ($x > 0$) and justify your answer.