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.

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.

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.

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 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 $C'(x)$.

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.

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 $f$.

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.

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.

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.

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.