Differentiation Rules and Derivatives of Standard Functions
A video you posted has $V(t) = 500e^{0.3t}$ views after $t$ hours. How fast are views coming in at hour 4? You could compute a limit of difference quotients every time, but nobody does. In this section you build a toolkit of rules that turns "find the derivative" into a quick, mechanical job, and you learn how to combine the rules when a function mixes several operations.
Power Rule, Sums and Roots
The power rule works for any exponent, so first rewrite roots and reciprocals as powers: $\sqrt{x} = x^{1/2}$, $\sqrt[3]{x} = x^{1/3}$, $\dfrac{1}{x^2} = x^{-2}$.
The derivative at a point is the slope of the tangent line there. For $f(x) = x^2$ the power rule gives $f'(x) = 2x$, so the slope at $x = 1$ is $2$: the tangent rises $2$ units for every $1$ unit to the right, as in the figure.
Find the derivative of each of the following functions using differentiation rules for elementary functions (powers and roots): a) $f(x) = x^7 - 5x^3 + \pi$ b)…
Find the derivative of each of the following functions using differentiation rules for elementary functions (powers and roots):
- a) $f(x) = x^7 - 5x^3 + \pi$
- b) $g(x) = \sqrt{x} + \frac{1}{x^2}$
Given the function $f(x) = \frac{x^4}{4} - \sqrt[3]{x}$, find its derivative function $f'(x)$ and simplify the result by expressing the root in radical form again.
Given the function $f(x) = \frac{x^4}{4} - \sqrt[3]{x}$, find its derivative function $f'(x)$ and simplify the result by expressing the root in radical form again.
For the quadratic function $f(x) = x^2 - 6x + 5$: a) Find the derivative $f'(x)$. b) Find the slope of the tangent line to the graph of the function at the point with…
For the quadratic function $f(x) = x^2 - 6x + 5$:
- a) Find the derivative $f'(x)$.
- b) Find the slope of the tangent line to the graph of the function at the point with x-coordinate $x = 4$.
Derivatives of Standard Functions
The blue curve is $2^x$ and the orange curve is its derivative $2^x \ln 2 \approx 0.69 \cdot 2^x$. Notice that the derivative has the same shape as the function, just scaled.
Find the derivative of the following transcendental functions using standard derivative rules: a) $f(x) = 4 \cdot \ln(x) + \log_{2}(x)$ b) $g(x) = 3^x - 5 \cdot \sin(x)$
Find the derivative of the following transcendental functions using standard derivative rules:
- a) $f(x) = 4 \cdot \ln(x) + \log_{2}(x)$
- b) $g(x) = 3^x - 5 \cdot \sin(x)$
A spring oscillates following simple harmonic motion described by the function $y(x) = 5 \cdot \cos(x) + 2$, where $y$ is the position in inches and $x$ is the time.…
A spring oscillates following simple harmonic motion described by the function $y(x) = 5 \cdot \cos(x) + 2$, where $y$ is the position in inches and $x$ is the time. Find the expression for the derivative function $y'(x)$, which represents the velocity of oscillation of the spring.
Find the values of the parameters $a, b \in \mathbb{R}$ so that the function $f(x) = a \cdot \sin(x) + b \cdot e^{-x}$ simultaneously satisfies the following conditions:…
Find the values of the parameters $a, b \in \mathbb{R}$ so that the function $f(x) = a \cdot \sin(x) + b \cdot e^{-x}$ simultaneously satisfies the following conditions:
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Passes through the point $(0, 3)$.
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Its derivative at the origin is $f'(0) = 2$.
The Chain Rule
Most functions are built by plugging one function into another: $\cos(x^2)$, $e^{3x}$, $(2x+1)^5$. The chain rule handles every one of them.
With $u = u(x)$, the standard table becomes:
$$(u^n)' = n u^{n-1}u' \quad (e^u)' = e^u u' \quad (\ln u)' = \dfrac{u'}{u} \quad (\sin u)' = \cos u \cdot u' \quad (\cos u)' = -\sin u \cdot u' \quad (\arctan u)' = \dfrac{u'}{1+u^2}$$Here the orange curve is the derivative of $\sin(x^2)$. It oscillates faster as $|x|$ grows, because the inside function $x^2$ changes faster and faster. That extra factor $2x$ is exactly what the chain rule adds.
Find the derivative of the following simple composite functions using the chain rule directly: a) $f(x) = (2x + 3)^5$ b) $g(x) = e^{x^2 - 1}$
Find the derivative of the following simple composite functions using the chain rule directly:
- a) $f(x) = (2x + 3)^5$
- b) $g(x) = e^{x^2 - 1}$
Find the derivative of the following composite function, applying the chain rule and simplifying the result as much as possible: $f(x) = \cos^3(e^{x^2+1})$
Find the derivative of the following composite function, applying the chain rule and simplifying the result as much as possible:
$f(x) = \cos^3(e^{x^2+1})$
Find the equation of the tangent line to the curve $f(x) = \arctan\left(\frac{x}{2}\right)$ at the point with x-coordinate $x = 2$.
Find the equation of the tangent line to the curve $f(x) = \arctan\left(\frac{x}{2}\right)$ at the point with x-coordinate $x = 2$.
Product and Quotient Rules
The strategy is always the same: find the outermost operation (a product, a quotient, a power...), apply its rule, and use the chain rule inside each derivative.
When you set a derivative equal to $0$, the equation can mix a logarithm with a rational expression. Then there may be no algebraic solution. Sometimes the honest answer is "never", and you can show it by comparing the possible values (the range) of the two sides.
A surveillance drone follows a trajectory whose height in feet is given by the function $h(t) = 15t \cdot e^{-0.2t}$, where $t$ is the time in seconds since takeoff. a)…
A surveillance drone follows a trajectory whose height in feet is given by the function $h(t) = 15t \cdot e^{-0.2t}$, where $t$ is the time in seconds since takeoff.
- a) Find the function that represents the vertical velocity of the drone, $v(t) = h'(t)$.
- b) What is the vertical velocity of the drone at $t = 5$ seconds?
Given the rational function involving natural logarithms: $g(x) = \frac{\ln(x^2 + 4)}{x}$ Find the expression for its derivative $g'(x)$ and determine the values of $x$…
Given the rational function involving natural logarithms:
$g(x) = \frac{\ln(x^2 + 4)}{x}$
Find the expression for its derivative $g'(x)$ and determine the values of $x$ for which the slope of the tangent line to the graph of $g(x)$ is equal to zero.
Simplify Before You Differentiate
Some functions look scary but collapse to something simple if you use logarithm properties or trig identities first. This saves a whole chain rule and many chances for sign errors.
Trig functions work the same way. When you see $1 \pm \sin x$ in a fraction, remember that $(1 - \sin x)(1 + \sin x) = 1 - \sin^2 x = \cos^2 x$. After differentiating, combine fractions over a common denominator and look for that pattern. The final answer is often surprisingly short, so always ask whether your result can be written as one standard function.
Find the derivative of the following function, simplifying the final result as much as possible using trigonometric identities:…
Find the derivative of the following function, simplifying the final result as much as possible using trigonometric identities:
$f(x) = \ln\left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right)$
Calculate the derivative of the following composite function and simplify the result as much as possible using trigonometric identities: $$g(x) = \ln \left(…
Calculate the derivative of the following composite function and simplify the result as much as possible using trigonometric identities:
$$g(x) = \ln \left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right)$$What do you notice about the final simplified result?
Logarithmic Differentiation
What about $x^{\sqrt{x}}$ or $(\sin x)^x$? The power rule needs a constant exponent and the exponential rule needs a constant base. When both the base and the exponent depend on $x$, no row of the table works directly.
Logarithmic differentiation is also handy for a long product or quotient such as $\dfrac{x^3\sqrt{x+1}}{(x-2)^4}$: take $\ln$ first and the product rule disappears.
Use the technique of logarithmic differentiation to obtain the derivative of the following power-exponential function: $f(x) = (x^2 + 1)^{\sin(x)}$
Use the technique of logarithmic differentiation to obtain the derivative of the following power-exponential function:
$f(x) = (x^2 + 1)^{\sin(x)}$
Given the transcendental function defined by $f(x) = (\arctan x)^x$ for $x > 0$, use the technique of logarithmic differentiation to find its derivative $f'(x)$. Once…
Given the transcendental function defined by $f(x) = (\arctan x)^x$ for $x > 0$, use the technique of logarithmic differentiation to find its derivative $f'(x)$. Once obtained, calculate the exact value of the slope of the tangent line to the graph of $f$ at the point with x-coordinate $x = 1$.
Implicit Differentiation and Normal Lines
Some curves, like a circle, cannot be written as a single function $y = f(x)$. You can still find slopes: treat $y$ as an unknown function of $x$ and differentiate both sides of the equation.
The normal line to a curve at a point is the line through that point that is perpendicular to the tangent. If the tangent has slope $m \ne 0$, the normal has slope $-\dfrac{1}{m}$.
For the circle $x^2 + y^2 = 25$: $2x + 2y\,y' = 0$, so $y' = -\dfrac{x}{y}$. At $(3, 4)$ the tangent slope is $-\dfrac34$, so the normal slope is $\dfrac43$. The normal passes through the center, as every normal to a circle does.
Consider the curve defined implicitly by the equation: $$x^2 \cdot y + y^3 = 10$$ a) Find the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit…
Consider the curve defined implicitly by the equation:
$$x^2 \cdot y + y^3 = 10$$- a) Find the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit differentiation.
- b) Find the equation of the normal line to the curve at the point $P(3, 1)$.
Derivative of an Inverse Function
A function that is strictly increasing (or strictly decreasing) on its whole domain is one-to-one, so it has an inverse. You can prove that with the derivative: if $f'(x) > 0$ for all $x$, then $f$ is strictly increasing. Some problems call a one-to-one function on all of $\mathbb{R}$ bijective.
Let the function be $f(x) = x^3 + 2x + \cos(x - 1)$. a) Prove that $f(x)$ is bijective on all of $\mathbb{R}$ (you can rely on the study of its increasing behavior). b)…
Let the function be $f(x) = x^3 + 2x + \cos(x - 1)$.
- a) Prove that $f(x)$ is bijective on all of $\mathbb{R}$ (you can rely on the study of its increasing behavior).
- b) Knowing that $f(1) = 4$, calculate the value of the derivative of the inverse function at that point, that is, $(f^{-1})'(4)$.
Differentiability of Piecewise Functions
Determine the values of the real parameters $a$ and $b$ so that the following piecewise function is differentiable throughout its domain: $$f(x) = \begin{cases} a \cdot…
Determine the values of the real parameters $a$ and $b$ so that the following piecewise function is differentiable throughout its domain:
$$f(x) = \begin{cases} a \cdot e^{x^2-1} + b & \text{if } x \leq 1 \\ \ln(x) + x^2 & \text{if } x > 1 \end{cases}$$Explain with reasoning the relationship between continuity and differentiability at the transition point $x = 1$.
Higher-Order Derivatives
The derivative of $f'$ is the second derivative $f''(x)$, and so on: $f^{(n)}(x)$ is the nth derivative. Just keep applying the rules, simplifying as you go.
To find the nth derivative of a rational function, first break it into partial fractions, then use the pattern for $(x + c)^{-1}$:
$$\dfrac{d}{dx}(x+c)^{-1} = -(x+c)^{-2},\quad \dfrac{d^2}{dx^2}(x+c)^{-1} = 2(x+c)^{-3},\quad \dfrac{d^3}{dx^3}(x+c)^{-1} = -6(x+c)^{-4}$$ $$\dfrac{d^n}{dx^n}\,(x+c)^{-1} = \dfrac{(-1)^n\, n!}{(x+c)^{n+1}}$$Given the function $f(x) = \ln(\cos(x))$, find: a) The first derivative $f'(x)$. b) The second derivative $f''(x)$, simplifying the result using standard trigonometric…
Given the function $f(x) = \ln(\cos(x))$, find:
- a) The first derivative $f'(x)$.
- b) The second derivative $f''(x)$, simplifying the result using standard trigonometric functions.
Find the general expression of the nth derivative ($f^{(n)}(x)$) for the rational function: $$f(x) = \frac{1}{x^2 - 3x + 2}$$ Hint: Decompose the function using partial…
Find the general expression of the nth derivative ($f^{(n)}(x)$) for the rational function:
$$f(x) = \frac{1}{x^2 - 3x + 2}$$Hint: Decompose the function using partial fractions before taking successive derivatives.
Derivatives as Rates of Change
The derivative $f'(t)$ is the instantaneous rate of change of $f$. Its units are (units of the output) per (unit of the input): feet per second for a height in feet, followers per month for a follower count.
The blue curve is the height $h(t) = 48t - 16t^2$ (in feet) of a ball thrown upward, and the orange line is its velocity $h'(t) = 48 - 32t$ in ft/s. At the top, $t = 1.5$ s, the velocity is $0$.
Back to the start: for $V(t) = 500e^{0.3t}$ the chain rule gives $V'(t) = 150e^{0.3t}$. At $t = 4$, $V'(4) = 150e^{1.2} \approx 498$ views per hour.
Sometimes two quantities change at once and are linked by a formula. Differentiate the formula with respect to time $t$ and use the chain rule (this is called a related rates problem).
A surveillance drone takes off vertically from a platform. The height $h$ in feet with respect to time $t$ in seconds is given by the polynomial function…
A surveillance drone takes off vertically from a platform. The height $h$ in feet with respect to time $t$ in seconds is given by the polynomial function $h(t) = t^3 - 4t^2 + 5t + 2$. Find the derivative function $h'(t)$, which represents the instantaneous velocity of the drone at any time $t$.
A social media account has analyzed its growth and determined that the number of followers $S$ evolves according to the exponential function $S(t) = 150 \cdot e^{0,4t}$,…
A social media account has analyzed its growth and determined that the number of followers $S$ evolves according to the exponential function $S(t) = 150 \cdot e^{0,4t}$, where $t$ is the time elapsed in months since its creation. Determine the derivative function $S'(t)$ to find the growth rate of followers.
In a biotechnology laboratory, the volume of a population of bacteria (in $mm^3$) grows according to the function $V(t) = 20 \cdot e^{0.5t}$, where $t$ is the time in…
In a biotechnology laboratory, the volume of a population of bacteria (in $mm^3$) grows according to the function $V(t) = 20 \cdot e^{0.5t}$, where $t$ is the time in hours. At the same time, the radius of the circular culture plate where the bacteria grow expands so that the area occupied by the population is $A(t) = \pi \cdot [r(t)]^2$.
If we assume that the thickness of the bacterial layer is constant and equal to $0.1 mm$ (so that $V = A \cdot h$), find the instantaneous rate of change of the radius of the population at $t = 2$ hours.
Minimizing a Distance
To find the point of a curve closest to a given point, minimize the distance $d$. Since $d \ge 0$, the square $d^2$ is smallest exactly where $d$ is smallest. Minimizing $d^2$ avoids a square root and an extra chain rule.
A surveillance drone moves following the trajectory given by the function $f(x) = \ln(x^2 + 1)$. An operator located at the point $A(2, 0)$ observes the drone. Find the…
A surveillance drone moves following the trajectory given by the function $f(x) = \ln(x^2 + 1)$. An operator located at the point $A(2, 0)$ observes the drone.
Find the point on the trajectory where the drone is at the minimum distance from the operator.
Note: To minimize the distance $d$, it is sufficient to minimize the square of the distance $d^2$. You will need to differentiate the resulting distance function.