Maths - Differentiation

Created: October 06, 2020
Updated: August 25, 2024

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Flashcards

What is the derivative of $x^n$?
\[nx^{(n-1)}\]

What is the derivative of $ax^n$?
\[anx^{(n-1)}\]

What is $f’(x)$ where $f(x) = 4x^2$?
\[8x\]

What is $\frac{dy}{dx}$ for $y = \frac{1}{2} x^{-4}$?
\[-2x^{-5}\]

What is the derivative of $x^{\frac{1}{2}}$?
\[-\frac{1}{2} x^{-\frac{1}{2}} \equiv -\frac{1}{2x^{\frac{3}{2}}}\]

Differentiating a polynomial with a highest power $n$ means the power becomes?

$n-1$.

What is the sum rule for differentiation?
\[h'(x) = af'(x) + bg'(x)\]

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?
\[0\]

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?
\[(x, f(x)) \to (x+h, f(x+h))\]
\[f(x) = x^2 - x + 1\]

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?
\[1\]

What is the limit definition of a derivative?
\[f'(x) = \lim_{x \to 0} \frac{f(x + h) - f(x)}{h}\]

What is the 3 step process for finding a derivative from first principles?
  1. Write out the general coordinates for $x$ and $x + h$
  2. Find an expression for the gradient
  3. 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?
\[h\]

PHOTO TANGENT CURVE LESS ACCURATE Visually, how could you make this approximation of the gradient of the curve more accurate?

PHOTO TANGENT CURVE MORE ACCURATE

2021-01-11

What is the equation for the tangent to a curve $y = f(x)$ at point $(a, f(a))$?
\[y - f(a) = f'(a)(x-a)\]

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?
\[-\frac{1}{f'(a)}\]

What is the equation for the normal to a curve $y = f(x)$ at point $(a, f(a))$?
\[y - f(a) = -\frac{1}{f'(a)}(x-a)\]

What is the derivative of the general quadratic $ax^2 + bx + c$?
\[2ax + b\]

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 $f(x)$?
\[f'(x)\]

What is the notation for the gradient of $y = …$?
\[\frac{dy}{dx}\]

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?
\[f''(x)\]

What is the $\frac{dy}{dx}$ notation for a second order derivative?
\[\frac{d^2y}{dx^2}\]

If the displacement of something is modelled as the function $f(x)$, what is the function for the acceleration?
\[f''(x)\]

What is true about a stationary point $x$ on a function $f(x)$?
\[f'(x) = 0\]

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.

PHOTO QUADRATIC TURNING POINT This a turning point of a quadratic. Is it a local maximum, local minimum or a point of inflection?

A local minimum.

PHOTO QUADRATIC TURNING POINT This a turning point of a quadratic. Is the gradient positive or negative to the left of the highlighted point?

Negative.

PHOTO QUADRATIC TURNING POINT This a turning point of a quadratic. Is the gradient positive or negative to the right of the highlighted point?

Positive.

PHOTO QUADRATIC TURNING POINT This a turning point of a quadratic. What is the value of $f’(x)$ at this point?
\[0\]

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 $t$?
\[\frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}\]

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 $y$?
\[y\]

What is $\frac{\text{d}}{\text{d}x} f(y)$ (respect to $x$!)?
\[f'(y)\frac{\text{d}y}{\text{d}x}\]

What is $\frac{\text{d}}{\text{d}x} y^m$ (respect to $x$!)?
\[my^{m-1} \frac{\text{d}y}{\text{d}x}\]

What is $\frac{\text{d}}{\text{d}x} xy$ (respect to $x$!)?
\[x \frac{\text{d}y}{\text{d}x} + y\]

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?
\[\frac{\text{d}y}{\text{d}x} \ne 0\] \[\frac{\text{d}^2y}{\text{d}x^2} = 0\]

What is a point of inflection that is also a stationary point called?

A stationary point of inflection.



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