Maths - Differentiation

Created: October 06, 2020 | Updated: August 25, 2024 | Read markdown | About these notes | View in context | Study these flashcards |

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The power rule

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

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

What is $f’(x)$ where $f(x) = 4x^2$?
derivative-of-four-x-squared
\[8x\]

What is $\frac{dy}{dx}$ for $y = \frac{1}{2} x^{-4}$?
derivative-of-half-x-to-minus-four
\[-2x^{-5}\]

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

Differentiating a polynomial with a highest power $n$ means the power becomes?
power-decreases-by-one

$n-1$.

The sum rule

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

What does the sum rule mean in practical terms?
sum-rule-term-by-term

That you can take the derivative of each term in a series, add them together and you get the derivative for the whole expression.

Tangents and turning points

What is the gradient of a tangent to a point on a curve the same as?
tangent-gradient-equals-curve-gradient

The gradient of the curve at the point.

What is the gradient of the local maximum and minimum points of a curve equal to?
turning-point-gradient-zero
\[0\]

Derivatives from First Principles

To approximate the gradient of a curve at a point, what can you do?
approximate-gradient-with-chord

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?
accuracy-by-closer-points

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

Expand the point $(x+h, f(x+h))$?
expand-point-x-plus-h
\[(x+h, (x+h)^2 - (1 + h) + 1)\]

What is the limit of $h + 1$ as $h$ approaches 0?
limit-of-h-plus-one
\[1\]

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

What is the 3 step process for finding a derivative from first principles?
first-principles-three-steps
  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?
first-principles-shrinking-variable
\[h\]

Visually, how could you make this approximation of the gradient of the curve more accurate?
make-approximation-more-accurate-visual

Tangents and normals

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

What is the normal to a curve at point $A$?
normal-definition

The line perpendicular to the tangent on the curve at $A$.

For a gradient $f’(a)$, what is the gradient for a line perpendicular to that point?
perpendicular-gradient-negative-reciprocal
\[-\frac{1}{f'(a)}\]

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

Differentiating the general quadratic

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

What is special about where the derivative of $ax^2 + bx + c$ crosses the x-axis?
quadratic-derivative-root-is-turning-point

It is the turning point of the quadratic.

Derivative notation

What is the notation for the gradient of $f(x)$?
lagrange-notation-for-f
\[f'(x)\]

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

Increasing and decreasing functions

What’s the difference between $f’(x)$ increasing and strictly increasing?
increasing-versus-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 $[a,b]$?
increasing-on-interval-definition

$f’(x) > 0$ for all x $a < x < b$.

What does it mean for $f(x)$ to be decreasing in the interval $[a,b]$?
decreasing-on-interval-definition

$f’(x) < 0$ for all x $a < x < b$.

When stating something is increasing or decreasing on an interval, what must you remember to do?
check-function-defined-on-bounds

Check that the function is defined for the bounds of the interval.

Second derivatives

What is the $f(x)$ notation for a second order derivative?
second-derivative-lagrange-notation
\[f''(x)\]

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

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

Classifying stationary points

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

What are the three types of stationary points called?
three-types-of-stationary-point
  • 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?
gradient-left-of-maximum

Positive.

Is $f’(x + h)$ for a local maximum $x$ and a small positive value $h$ positive or negative?
gradient-right-of-maximum

Negative.

Is $f’(x - h)$ for a local minimum $x$ and a small positive value $h$ positive or negative?
gradient-left-of-minimum

Negative.

Is $f’(x + h)$ for a local minimum $x$ and a small positive value $h$ positive or negative?
gradient-right-of-minimum

Positive.

If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) > 0$?
second-derivative-positive-means-minimum

The point is a local minimum.

If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) < 0$?
second-derivative-negative-means-maximum

The point is a local maximum.

If $f(x)$ has a stationary point $x = a$, what does it mean if $f’‘(a) = 0$?
second-derivative-zero-is-inconclusive

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?
same-sign-either-side-is-inflection

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?
quadratic-turning-point-is-minimum

A local minimum.

This a turning point of a quadratic. Is the gradient positive or negative to the left of the highlighted point?
quadratic-gradient-left-of-minimum

Negative.

This a turning point of a quadratic. Is the gradient positive or negative to the right of the highlighted point?
quadratic-gradient-right-of-minimum

Positive.

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

Derivatives of exponentials and logarithms

\[f(x) = e^x\]

What is $f’(x)$?
derivative-of-e-to-the-x
\[e^x\]
\[f(x) = \ln x\]

What is $f’(x)$?
derivative-of-natural-log
\[\frac{1}{x}\]
\[y = \ln 2x\]

How could you rewrite this in order to find the derivative?
rewrite-log-of-2x-as-sum
\[y = \ln 2 + \ln x\]
\[y = \ln 2 + \ln x\]

What is the derivative?
derivative-of-ln-2-plus-ln-x
\[\frac{1}{x}\]
\[y = 5\ln x\]

What is the derivative?
derivative-of-five-ln-x
\[\frac{5}{x}\]
\[\ln ax\]

Why is the derivative always $\frac{1}{x}$?
why-derivative-of-ln-ax-is-one-over-x

Because you could rewrite it as $\ln a + \ln x$ and the constant term would disappear.

Derivatives of trigonometric functions

\[\frac{d}{dx}(\cos x)\]

What is this?
derivative-of-cos
\[-\sin x\]
\[\frac{d}{dx}(\sin x)\]

What is this?
derivative-of-sin
\[\cos x\]
\[\frac{d}{dx}(\tan x)\]

What is this?
derivative-of-tan
\[\sec^2 x\]
\[\frac{d}{dx}(\csc x)\]

What is this?
derivative-of-csc
\[-\csc x\cot x\]
\[\frac{d}{dx}(\sec x)\]

What is this?
derivative-of-sec
\[\sec x\tan x\]
\[\frac{d}{dx}(\cot x)\]

What is this?
derivative-of-cot
\[-\csc^2 x\]

Derivatives of general exponentials

What is the derivative of

\[y = a^{kx}\]

?
derivative-of-a-to-the-kx
\[k\ln a \times a^{kx}\]

What is the derivative of

\[y = 3^{4x}\]

?
derivative-of-three-to-the-4x
\[4\ln 3 \times 3^{4x}\]

What is the derivative of

\[y = \frac{3}{2}^{2x}\]

?
derivative-of-three-halves-to-the-2x
\[2\ln\left(\frac{3}{2}\right) \times \left(\frac{3}{2}\right)^{2x}\]

Parametric differentiation

What is the derivative $\frac{\text{d}y}{\text{d}x}$ in terms of a parameter $t$?
parametric-derivative-formula
\[\frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}\]

If

\[x = 2t\] \[y = t^2 - 3t + 2\]

how could you find $\frac{\text{d}x}{\text{d}y}$?
parametric-example-2t-and-quadratic
\[\frac{2t-3}{2}\]

When differentiating parametrically, what goes on top, $x$ or $y$?
parametric-y-on-top
\[y\]

Implicit differentiation

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

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

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

A worked chain-rule derivative

\[y = \ln (3x) - e^{-2x}\]

What is the derivative of this (you’ve got this wrong twice)?
derivative-of-ln-3x-minus-e-to-minus-2x
\[\frac{1}{x} + 2e^{-2}x\]

Points of inflection

Is a point of inflection necessarily a stationary point?
inflection-need-not-be-stationary

No.

If a point of inflection isn’t a stationary point, what will be true in terms of derivatives?
non-stationary-inflection-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?
stationary-point-of-inflection

A stationary point of inflection.



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