Maths - Differentiation

Flashcards

$$ nx^{(n-1)} $$

$$ anx^{(n-1)} $$

$$ -2x^{-5} $$

$$ -\frac{1}{2} x^{-\frac{1}{2}} \equiv -\frac{1}{2x^{\frac{3}{2}}} $$

$$ h'(x) = af'(x) + bg'(x) $$

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

The gradient of the curve at the point.

Draw a line through the point and another point nearby on the curve.

Move the two points closer together.

$$ (x, f(x)) \to (x+h, f(x+h)) $$

$$ (x+h, (x+h)^2 - (1 + h) + 1) $$

$$ f'(x) = \lim_{x \to 0} \frac{f(x + h) - f(x)}{h} $$

Write out the general coordinates for $x$ and $x + h$
Find an expression for the gradient
See what expression becomes as $h$ approaches zero.

PHOTO TANGENT CURVE MORE ACCURATE

$$ y - f(a) = f'(a)(x-a) $$

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

$$ -\frac{1}{f'(a)} $$

$$ y - f(a) = -\frac{1}{f'(a)}(x-a) $$

$$ 2ax + b $$

It is the turning point of the quadratic.

$$ \frac{dy}{dx} $$

Increasing: $f'(x) \ge 0$
Strictly increasing: $f'(x) > 0$

$f'(x) > 0$ for all x $a < x < b$.

$f'(x) < 0$ for all x $a < x < b$.

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

$$ f''(x) $$

$$ \frac{d^2y}{dx^2} $$

$$ f''(x) $$

$$ f'(x) = 0 $$

Local maxima
Local minima
Point of inflection

The point is a local minimum.

The point is a local maximum.

The point could be a local minimum, local maximum or a point of inflection.

It is a point of inflection.

A local minimum.

$$ \frac{1}{x} $$

$$ y = \ln 2 + \ln x $$

$$ \frac{1}{x} $$

$$ \frac{5}{x} $$

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

$$ -\sin x $$

$$ \cos x $$

$$ \sec^2 x $$

$$ -\csc x\cot x $$

$$ \sec x\tan x $$

$$ -\csc^2 x $$

$$ k\ln a \times a^{kx} $$

$$ 4\ln 3 \times 3^{4x} $$

$$ 2\ln\left(\frac{3}{2}\right) \times \left(\frac{3}{2}\right)^{2x} $$

$$ \frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}} $$

$$ \frac{2t-3}{2} $$

$$ f'(y)\frac{\text{d}y}{\text{d}x} $$

$$ my^{m-1} \frac{\text{d}y}{\text{d}x} $$

$$ x \frac{\text{d}y}{\text{d}x} + y $$

Use the chain rule but backwards

$$ \frac{\text{d}A}{\text{d}r} \times \frac{\text{d}r}{\text{d}t} = \frac{\text{d}A}{\text{d}t} $$

date: 2020-10-06 18:31
tags:
- '@?maths'
- '@?further-maths'
- '@?public'
title: Maths - Differentiation

quadratic-turning-point.png

Maths - Differentiation

See Also

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??

What is the sum rule for differentiation??

What does the sum rule mean in practical terms??

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

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??

How could you get a more accurate value of the gradient by drawing a line through two points on a curve??

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 $(x+h, f(x+h))$??

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??

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??

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 $A$??

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??

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??

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

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

When stating something is increasing or decreasing on an interval, what must you remember to do??

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??

Is $f'(x - h)$ for a local maximum $x$ and a small positive value $h$ positive or negative??

Is $f'(x + h)$ for a local maximum $x$ and a small positive value $h$ positive or negative??

Is $f'(x - h)$ for a local minimum $x$ and a small positive value $h$ positive or negative??

Is $f'(x + h)$ for a local minimum $x$ and a small positive value $h$ positive or negative??

If $f(x)$ has a stationary point $x = a$, what does it mean if $f''(a) > 0$??

If $f(x)$ has a stationary point $x = a$, what does it mean if $f''(a) < 0$??

If $f(x)$ has a stationary point $x = a$, what does it mean if $f''(a) = 0$??

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??

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 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 right of the highlighted 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 $f'(x)$??

$$f(x) = \ln x$$ What is $f'(x)$??

2021-02-03

$$y = \ln 2x$$ How could you rewrite this in order to find the derivative??

$$y = \ln 2 + \ln x$$ What is the derivative??

$$y = 5\ln x$$ What is the derivative??

$$\ln ax$$ Why is the derivative always $\frac{1}{x}$??

2021-02-11

$$\frac{d}{dx}(\cos x)$$ What is this??

$$\frac{d}{dx}(\sin x)$$ What is this??

$$\frac{d}{dx}(\tan x)$$ What is this??

$$\frac{d}{dx}(\csc x)$$ What is this??

$$\frac{d}{dx}(\sec x)$$ What is this??

$$\frac{d}{dx}(\cot x)$$ What is this??

2021-07-31

What is the derivative of $$y = a^{kx}$$??

What is the derivative of $$y = 3^{4x}$$??

What is the derivative of $$y = \frac{3}{2}^{2x}$$??

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 $\frac{\text{d}x}{\text{d}y}$??

When differentiating parametrically, what goes on top, $x$ or $y$??

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