Introduction to Differentiation

calculus year11 advanced

Introduction to differentiation | NSW Curriculum Website

• MAV-11-06 interprets the meaning of the derivative and determines the derivative of functions to solve problems

📖 Prior Knowledge

Content	Prerequisite relationships
Ratios and Rates	- Interpret distance–time graphs → Find average speed from a graph
Algebraic Techniques A	- Expand binomial products → Differentiate from first principles
Variation and Rates of Change B	- Analyse graphs with a constant rate of change → Calculate an average rate of change
Indices C	- Describe and use fractional indices → Apply the power rule
Equations C	- Solve quadratic equations by factorisation → Find stationary points
Linear Relationships C	- Apply the gradient formula → Calculate an average rate of change - Find a line’s equation using point–gradient form → Find tangents and normals
Trigonometry D	- Relate gradient to angle of inclination → Relate tangent gradient to inclination
Working with Functions	- Form and evaluate composite functions → Apply the chain 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 $n{x}^{n-1}$ is the gradient of $f(x)=x^n$ and verify the result using graphing applications
• Define $f^{\prime }(x)$, 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 $f^{\prime }(x)$ 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 \displaystyle 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 $y^{\prime }$ for the derivative of $y$ when $y$ is a function of $x$
• Use the notation $\frac{d}{dx}(f(x))$ and $f^{\prime }(x)$ for the derivative of a function $f(x)$
• Use the formula $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for all real values of $n$
• Apply the fact that the derivative of a sum is the sum of the derivatives: $\frac{d}{dx}(f(x)+g(x))=f^{\prime }(x)+g^{\prime }(x)$, and the derivative of a multiple of a function is the multiple of its derivative: $\frac{d}{dx}(k\cdot f(x))=k\cdot f^{\prime }(x)$
• 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 $h^{\prime }(x)=f(x)g^{\prime }(x)+f^{\prime }(x)g(x)$
• 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 $h^{\prime }(x)=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$
• 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 $h^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x)$
• 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 $f^{\prime }(c)>0$ and decreasing at $x=c$ when $f^{\prime }(c)<0$
• 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 $f^{\prime }(c)=0$
• Graph $y=f^{\prime }(x)$ 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 $f^{\prime }(c)$ 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