Introduction to Differentiation

#Calculus #Year11 #Advanced >[!info]- [Introduction to differentiation | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n11/fa6bc41678) >- MAV-11-06 interprets the meaning of the derivative and determines the derivative of functions to solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ----------------------------------- | ----------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Variation and Rates of Change B]] | - rates of change | - concept of derivative representing the rate of change | | [[Working with Functions]] | - function notation - linear functions - non-linear functions - index laws | - working with derivatives $f'(x)lt;br>- tangents and normals are linear functions - sketching non-linear functions to interpret a derivative graphically - power rule | ## Estimating change - Define the average rate of change of $y$ with respect to $x$ for a function $y=f(x)$ over the domain $[a,b]$ as $\frac{\Delta y}{\Delta x}=\frac{\text{change in } y}{\text{change in }x}$, that is $\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$, and recognise $\frac{f(b)-f(a)}{b-a}$ as the gradient of the secant through $(a,f(a))$ and $(b,f(b))$ on the graph of $y=f(x)$ - Recognise speed as a rate of change of distance with respect to time - Use the definition for average rate of change to determine the average speed of an object from a given distance–time graph - Describe the difference between the average speed of an object and its instantaneous speed - Determine that the instantaneous speed of an object at time t can be approximated by the average speed between its position at time t and its position some time later, and explain how this approximation can be improved - Relate the instantaneous speed of an object to the gradient of the tangent at that point on its distance–time graph - Estimate the instantaneous speed of an object from its distance–time graph - Recognise when modelling with a linear function that its gradient is the rate of change and determine the rate of change for linear functions in practical situations - Recognise when modelling with a non-linear function that the rate of change is not constant and is represented by the gradient of the tangent to the curve at each point on the curve - Estimate the instantaneous rate of change of a non-linear function at a given point from a given graph of a practical situation ## The derivative - Examine the gradient of a curve at a point on the curve using graphing applications - Approximate the gradient of a curve $f(x)=x^n$ at a point $P(c,f(c))$ by considering the gradient of the secant through $P$ and $Q(c+h,f(c+h))$ as the magnitude of h approaches zero, using graphing applications or a spreadsheet - Infer that $nx^{n-1}$ is the gradient of $f(x)=x^n$ and verify the result using graphing applications - Define $f'(x)$, for any function $f(x)$ and any value x, to be the gradient of the tangent to the curve $y=f(x)$ at the point $P(x,f(x))$ if the tangent exists and is not vertical - Refer to $f'(x)$ as the derivative of $f(x)$ or the gradient, or derived, function of $f(x)$ - Define differentiation as the process of finding the derivative of a function - Find derivatives of constant and linear functions - Define the derivative of the function $f(x)$ from first principles, as the limiting value of the gradient of the secant $\frac{f(x+h)-f(x)}{h}$ as h approaches zero, when this limiting value exists, and use the notation $f'(x)=\lim_\limits{h\to0}\frac{f(x+h)-f(x)}{h}$ - Use first principles to find the derivative of quadratic functions ## Calculations with the derivative - Use the notation $\frac{dy}{dx}$ and $y'$ for the derivative of $y$ when $y$ is a function of $x$ - Use the notation $\frac{d}{dx}(f(x))$ and $f'(x)$ for the derivative of a function $f(x)$ - Use the formula $\frac{d}{dx}(x^n)=nx^{n-1}$ for all real values of $n$ - Apply the fact that the derivative of a sum is the sum of the derivatives: $\frac{d}{dx}(f(x)+g(x))=f'(x)+g'(x)$, and the derivative of a multiple of a function is the multiple of its derivative: $\frac{d}{dx}(k\cdot f(x))=k\cdot f'(x)$ - Use the rules for differentiation to find equations of tangents and normals to a curve at points on the curve - Find points on a curve where the tangent or normal has a given gradient - Examine and use the relationship between the angle of inclination of a line or tangent to a curve, $\theta$, with the positive x-axis, and the gradient, m, of that line or tangent, and establish that $\tan{\theta}=m$ - Apply the product rule: if $y=uv$, where u and v are both differentiable functions of x, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$ or if $h(x)=f(x)g(x)$ for differentiable functions $f(x)$ and $g(x)$ then $h'(x)=f(x)g'(x)+f'(x)g(x)$ - Apply the quotient rule: if $y=\frac{u}{v}$, where u and v are both functions of x, then $\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$ or if $h(x)=\frac{f(x)}{g(x)}$ then $h'(x)=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$ - Apply the chain rule: if y is a differentiable function of u, and u is a differentiable function of x, then $\frac{dy}{dx} = \frac{dy}{du}\times\frac{du}{dx}$ or if $h(x)=f(g(x))$ for differentiable functions $f(x)$ and $g(x)$ then $h'(x)=f'(g(x))g'(x)$ - Identify and apply the product, quotient or chain rule, or a combination of the rules, as appropriate to differentiate a given function ## Graphical applications of the derivative - Interpret $f(x)$ as increasing at $x=c$ when $f'(c)>0$ and decreasing at $x=c$ when $f'(c)<0$ - Describe the behaviour of a function at a point as stationary when the tangent at the point is parallel to the x-axis, and recognise that $f(x)$ is stationary at $x=c$ when $f'(c)=0$ - Graph $y=f'(x)$ for a given graph of a function $y=f(x)$ - Numerically estimate the value of the derivative at a point on the graph of a power of $x$, with and without the use of digital tools - Identify stationary points on the graph of a cubic function, and the values of $x$ for which the function is increasing and/or decreasing, by first calculating the derivative, and justify conclusions ## The derivative as a rate of change - Interpret $f'(c)$ as the instantaneous rate of change of the function $f(x)$ at $x=c$ - Define and distinguish between displacement and distance and between velocity and speed - Use graphs of functions and their derivatives, without the use of algebraic techniques, to describe and interpret physical phenomena - Use the notation $\frac{dx}{dt}$ or $\dot{x}$ to represent the velocity of a particle with displacement $x$ from a point as a function of time $t$ - Solve problems by determining the velocity of a particle moving in a straight line, given its displacement from a point as a function of time