Maths - Differentiation
See Also
- https://www.desmos.com/calculator/oyp1sie3gb
- [[Maths - Modelling with Differentiation]]A
- [[Maths - Sketching Gradient Functions]]A
- [[Maths - Chain Rule]]A
- [[Maths - Product Rule]]A
Flashcards
What is the derivative of $x^n$?
What is the derivative of $ax^n$?
What is $f’(x)$ where $f(x) = 4x^2$?
What is $\frac{dy}{dx}$ for $y = \frac{1}{2} x^{-4}$?
What is the derivative of $x^{\frac{1}{2}}$?
Differentiating a polynomial with a highest power $n$ means the power becomes?
$n-1$.
What is the sum rule for differentiation?
What does the sum rule mean in practical terms?
That you can take the derivative of each term in a series, add them together and you get the derivative for the whole expression.
What is the gradient of a tangent to a point on a curve the same as?
The gradient of the curve at the point.
What is the gradient of the local maximum and minimum points of a curve equal to?
Derivatives from First Principles
To approximate the gradient of a curve at a point, what can you do?
Draw a line through the point and another point nearby on the curve.
How could you get a more accurate value of the gradient by drawing a line through two points on a curve?
Move the two points closer together.
For a general function $f(x)$, what are the two points for finding the gradient $h$ away from the original point?
#####
\[f(x) = x^2 - x + 1\]Expand the point $(x+h, f(x+h))$??
\[(x+h, (x+h)^2 - (1 + h) + 1)\]What is the limit of $h + 1$ as $h$ approaches 0?
What is the limit definition of a derivative?
What is the 3 step process for finding a derivative from first principles?
- Write out the general coordinates for $x$ and $x + h$
- Find an expression for the gradient
- See what expression becomes as $h$ approaches zero.
When finding a derivative from first principles, what variable is used to represent a quantity that shrinks to zero?
Visually, how could you make this approximation of the gradient of the curve more accurate?
Visually, how could you make this approximation of the gradient of the curve more accurate?
2021-01-11
What is the equation for the tangent to a curve $y = f(x)$ at point $(a, f(a))$?
What is the normal to a curve at point $A$?
The line perpindicular to the tangent on the curve at $A$.
For a gradient $f’(a)$, what is the gradient for a line perpindicular to that point?
What is the equation for the normal to a curve $y = f(x)$ at point $(a, f(a))$?
What is the derivative of the general quadratic $ax^2 + bx + c$?
What is special about where the derivative of $ax^2 + bx + c$ crosses the x-axis?
It is the turning point of the quadratic.
2021-01-13
What is the notation for the gradient of $f(x)$?
What is the notation for the gradient of $y = …$?
2021-01-18
What’s the difference between $f’(x)$ increasing and strictly increasing?
- Increasing: $f’(x) \ge 0$
- Strictly increasing: $f’(x) > 0$
What does it mean for $f(x)$ to be increasing in the interval $[a,b]$?
$f’(x) > 0$ for all x $a < x < b$.
What does it mean for $f(x)$ to be decreasing in the interval $[a,b]$?
$f’(x) < 0$ for all x $a < x < b$.
When stating something is increasing or decreasing on an interval, what must you remember to do?
Check that the function is defined for the bounds of the interval.
2021-01-20
What is the $f(x)$ notation for a second order derivative?
What is the $\frac{dy}{dx}$ notation for a second order derivative?
If the displacement of something is modelled as the function $f(x)$, what is the function for the acceleration?
What is true about a stationary point $x$ on a function $f(x)$?
What are the three types of stationary points called?
- Local maxima
- Local minima
- Point of inflection
Is $f’(x - h)$ for a local maximum $x$ and a small positive value $h$ positive or negative?
Positive.
Is $f’(x + h)$ for a local maximum $x$ and a small positive value $h$ positive or negative?
Negative.
Is $f’(x - h)$ for a local minimum $x$ and a small positive value $h$ positive or negative?
Negative.
Is $f’(x + h)$ for a local minimum $x$ and a small positive value $h$ positive or negative?
Positive.
If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) > 0$?
The point is a local minimum.
If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) < 0$?
The point is a local maximum.
If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) = 0$?
The point could be a local minimum, local maximum or a point of inflection.
If $f’(x + h)$ and $f’(x - h)$ are the same a stationary point $x$ and a small positive value $h$, what must be true about the stationary point?
It is a point of inflection.
This a turning point of a quadratic. Is it a local maximum, local minimum or a point of inflection?
This a turning point of a quadratic. Is it a local maximum, local minimum or a point of inflection?A local minimum.
This a turning point of a quadratic. Is the gradient positive or negative to the left of the highlighted point?
This a turning point of a quadratic. Is the gradient positive or negative to the left of the highlighted point?Negative.
This a turning point of a quadratic. Is the gradient positive or negative to the right of the highlighted point?
This a turning point of a quadratic. Is the gradient positive or negative to the right of the highlighted point?Positive.
This a turning point of a quadratic. What is the value of $f’(x)$ at this point?
This a turning point of a quadratic. What is the value of $f’(x)$ at this point?2021-02-02
#####
\[f(x) = e^x\]What is $f’(x)$??
\[e^x\]#####
\[f(x) = \ln x\]What is $f’(x)$??
\[\frac{1}{x}\]2021-02-03
#####
\[y = \ln 2x\]How could you rewrite this in order to find the derivative??
\[y = \ln 2 + \ln x\]#####
\[y = \ln 2 + \ln x\]What is the derivative??
\[\frac{1}{x}\]#####
\[y = 5\ln x\]What is the derivative??
\[\frac{5}{x}\]#####
\[\ln ax\]Why is the derivative always $\frac{1}{x}$?? Because you could rewrite it as $\ln a + \ln x$ and the constant term would dissapear.
2021-02-11
#####
\[\frac{d}{dx}(\cos x)\]What is this??
\[-\sin x\]#####
\[\frac{d}{dx}(\sin x)\]What is this??
\[\cos x\]#####
\[\frac{d}{dx}(\tan x)\]What is this??
\[\sec^2 x\]#####
\[\frac{d}{dx}(\csc x)\]What is this??
\[-\csc x\cot x\]#####
\[\frac{d}{dx}(\sec x)\]What is this??
\[\sec x\tan x\]#####
\[\frac{d}{dx}(\cot x)\]What is this??
\[-\csc^2 x\]2021-07-31
What is the derivative of
\[y = a^{kx}\]??
\[k\ln a \times a^{kx}\]What is the derivative of
\[y = 3^{4x}\]??
\[4\ln 3 \times 3^{4x}\]What is the derivative of
\[y = \frac{3}{2}^{2x}\]??
\[2\ln\left(\frac{3}{2}\right) \times \left(\frac{3}{2}\right)^{2x}\]2021-12-15
What is the derivative $\frac{\text{d}y}{\text{d}x}$ in terms of a parameter $t$?
If
\[x = 2t\] \[y = t^2 - 3t + 2\]how could you find $\frac{\text{d}x}{\text{d}y}$??
\[\frac{2t-3}{2}\]When differentiating parametrically, what goes on top, $x$ or $y$?
What is $\frac{\text{d}}{\text{d}x} f(y)$ (respect to $x$!)?
What is $\frac{\text{d}}{\text{d}x} y^m$ (respect to $x$!)?
What is $\frac{\text{d}}{\text{d}x} xy$ (respect to $x$!)?
2022-02-03
If the rate of change of radius, $\frac{\text{d}r}{\text{d}t}$, remains at a constant $3$, and the rate of change of surface area is $\frac{\text{d}A}{\text{d}r} = 2\pi r$ then how could you find an expression for $\frac{\text{d}A}{\text{d}t}$?
Use the chain rule but backwards
\[\frac{\text{d}A}{\text{d}r} \times \frac{\text{d}r}{\text{d}t} = \frac{\text{d}A}{\text{d}t}\]2022-04-17
#####
\[y = \ln (3x) - e^{-2x}\]What is the derivative of this (you’ve got this wrong twice)??
\[\frac{1}{x} + 2e^{-2}x\]2022-04-30
Is a point of inflection necessarily a stationary point?
No.
If a point of inflection isn’t a stationary point, what will be true in terms of derivatives?
What is a point of inflection that is also a stationary point called?
A stationary point of inflection.