Derivative Exercises
Ximena is studying for her Calculus exam and runs into a function like $\dfrac{x^2}{x-3}$. She knows all the rules, but she doesn't know which one to start with or how to leave the result clean. On this page you will learn to look at a function and decide at a glance which rule takes charge, and how to simplify the result.
Deciding which rule to apply first
Rules we will use, with $u$ and $v$ functions of $x$:
| Function | Derivative |
|---|---|
| $x^n$ | $n x^{n-1}$ |
| $u^n$ | $n u^{n-1}\cdot u'$ |
| $u\cdot v$ | $u'v + uv'$ |
| $\dfrac{u}{v}$ | $\dfrac{u'v - uv'}{v^2}$ |
Determine the derivative of the rational function $f(x) = \frac{3x - 1}{x^2 + 4}$.
Determine the derivative of the rational function $f(x) = \frac{3x - 1}{x^2 + 4}$.
The second derivative
The derivative of a function is another function, so it can be differentiated too. The derivative of $f'$ is called the second derivative and is written $f''(x)$. You differentiate twice, one after the other, using the usual rules.
What is it good for? If $s(t)$ is position, then $s'$ is velocity and $s''$ is acceleration. On a graph, the sign of $f''$ tells you which way the curve bends: if $f''>0$ it is concave up, and if $f''<0$, concave down.
Calculate the first and second derivatives of the polynomial function $f(x) = 3x^4 - 5x^3 + 2x - 7$.
Calculate the first and second derivatives of the polynomial function $f(x) = 3x^4 - 5x^3 + 2x - 7$.
Chains and powers
When the function is an expression raised to a power, the chain rule applies: you differentiate from the outside in, layer by layer, without skipping any.
Calculate the derivative of the function $f(x) = (5x^3 - 2x)^4$ using the chain rule.
Calculate the derivative of the function $f(x) = (5x^3 - 2x)^4$ using the chain rule.
Simplifying the result
Back to the beginning: for $\dfrac{x^2}{x-3}$ the quotient rule takes charge, and the simplified result is $\dfrac{x(x-6)}{(x-3)^2}$.
Marginal cost
When a function describes a quantity that depends on production, its derivative is a rate of change. The marginal cost is the derivative of the total cost: it approximates how much it costs to produce one more item.
Extension: roots and fractional powers
A root can be seen as a power: $\sqrt{u}=u^{1/2}$ and $\sqrt[3]{x}=x^{1/3}$. With the chain rule we get the rule $\left(\sqrt{u}\right)'=\dfrac{u'}{2\sqrt{u}}$.
A cost with a root, like the one for the graduation-trip sweatshirts, is differentiated the same way: the marginal cost is the derivative of $C(x)$, and here the quotient and chain rules work together.
Find the derivative of the irrational function $f(x) = \sqrt{x^4 - 3x^2 + 2}$.
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.
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 $f(x) = \frac{\sqrt{x^2-1}}{x+2}$ and simplify the result as much as possible, expressing it with a single denominator.
Calculate the derivative of the 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 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…
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 $C'(x)$.
Extension: exponentials, logarithms, and trigonometric functions
Rules, with $u$ a function of $x$:
| Function | Derivative |
|---|---|
| $e^u$ | $e^u \cdot u'$ |
| $a^u$ | $a^u \ln a \cdot u'$ |
| $\ln u$ | $\dfrac{u'}{u}$ |
| $\sin u$ | $\cos u \cdot u'$ |
| $\cos u$ | $-\sin u \cdot u'$ |
| $\arcsin u$ | $\dfrac{u'}{\sqrt{1-u^2}}$ |
The game plan is the same: first the outer operation, then the inner layers.
Find the derivative of the function $f(x) = (2x^2 + 3) e^x$.
Find the derivative of the function $f(x) = (2x^2 + 3) e^x$.
Find the derivative of the trigonometric function $f(x) = \cos(x^3 - 4x)$.
Find the derivative of the trigonometric function $f(x) = \cos(x^3 - 4x)$.
Calculate the derivative of the function $f(x) = \ln(7x^2 + 5) + 3^{2x}$.
Calculate the derivative of the function $f(x) = \ln(7x^2 + 5) + 3^{2x}$.
Calculate the derivative of the composite function $f(x) = \sin^3(e^{2x} + 1)$, taking care with the successive application of the chain rule.
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 $x = \sqrt{\frac{\pi}{12}}$.
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…
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.
Extension: derivatives as rates in science
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…
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.
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 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 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 $t$…
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 $10^\circ \text{C/min}$.
Extension: challenges
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…
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 $n$.
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 $g'(2) = -4$,…
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…
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.
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 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 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.
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.