# Maths - Differentiation > Source: https://ollybritton.com/notes/a-level/maths/topics/differentiation/ · Updated: 2024-08-25 · Tags: maths, further-maths ## See Also - - [Maths - Modelling with Differentiation](https://ollybritton.com/notes/a-level/maths/topics/modelling-with-differentiation/) - [Maths - Sketching Gradient Functions](https://ollybritton.com/notes/a-level/maths/topics/sketching-gradient-functions/) - [Maths - Chain Rule](https://ollybritton.com/notes/a-level/maths/topics/chain-rule/) - [Maths - Product Rule](https://ollybritton.com/notes/a-level/maths/topics/product-rule/) ## 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](paste-3780cd4d12a1e96fc173afb3328efe17eb78176b.jpg) Visually, how could you make this approximation of the gradient of the curve more accurate?? ![PHOTO TANGENT CURVE MORE ACCURATE](paste-53c5bfe6beb2bf405d80805f5ac424fc72458af6.jpg) ### 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](quadratic-turning-point.png) 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](quadratic-turning-point.png) 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](quadratic-turning-point.png) 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](quadratic-turning-point.png) 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. --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt