14D - Differentiation of polynomial functions
For polynomial functions, the derivative can be determined using the following rule:
■ If \(f(x)=ax^n\), then \(f'(x)=n \times ax^{n-1}\).
For a constant term, \(k \in R \), the derivative is always zero. That is, if \(f(x)=k\), then \(f'(x)=0\).
14D - Example 1: Differentiation of polynomial functions by rule
Determine the derivative of each of the following polynomial functions.
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14D - Example 1: Video solution
14D - Example 1: Practice
Question 1: If \(f(x)=3x^2+7x-5\), find \(f'(x)\). Question 2: If \(y=5x-2x^4\), determine \(\frac{dy}{dx}\). Question 3: Compute \(\frac{d}{dx}(x^5)\). 14D - Example 1: Solutions
Question 1: \(f'(x)=6x+7\) Question 2: \(\frac{dy}{dx}=5-8x^3\) Question 3: \(\frac{d}{dx}(x^5)=5x^4\) |
14D - Example 2: Differentiation of polynomial functions by rule
Consider the graph of the function \(f(x)=x^2-4x\) shown.
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14D - Example 2: Video solution
14D - Example 2: Practice
Question 1: Consider the rule \(f(x)=2x^2-x^4\). Calculate the gradient of a tangent to the curve at \(x=-2\). Question 2: For the curve with rule \(y=\frac{x^2}{4}-3x\), determine the value of \(x\) such that the gradient of the tangent to the curve is \(-5\). Question 3: Consider the polynomial with rule \(h(x)=x^3-b^2x\), where \(b \in R\). Determine the value(s) of \(x\), where \(h'(x)=0\). State your answer in terms of \(b\). 14D - Example 2: Solutions
Question 1: \(f'(x)=-4x^3+4x\); therefore, \(f'(-2)=24\). Question 2: \(f'(x)=-5\); therefore, \(x=-4\). Question 3: \(x=\frac{-\sqrt{3}b}{3}\) and \(x=\frac{\sqrt{3}b}{3}\) |
14D - Example 3: Determining the equation of a tangent line (CAS)
Find the equation of the tangent line to the curve \(y=x^2-8x+16\) at the point \((3,1)\).
■ CAS Hint:The equation of a tangent line to the curve \(y=f(x)\) at \(x=a\) can be found using Interactive → Calc → Line → tanLine. ■ Note: Determining the equation of a straight line was covered in detail in Section 3B. |
14D - Example 3: Video solution
14D - Example 3: Practice
Question 1: Determine the equation of the tangent line to the curve \(f(x)=4-x^2\) at \(x=1\). Question 2: Determine the equation of the tangent line to the curve \(y=x^3+3x^2\) at the point \((-1,2)\). Question 3: The parabola \(y=x^2-7x+6\) has a tangent line with equation \(y=-11x+c\). Determine the coordinates of the point where the tangent touches the parabola and, hence, find the value of \(c\). Question 4: The cubic function \(g(x)=\frac{-x^3}{3}+x^2+5x\) has two tangent lines with a gradient of \(2\). Determine the equations for both of these tangent lines. 14D - Example 3: Solutions
Question 1: \(y=-2x+5\) Question 2: \(y=-3x-1\) Question 3: The tangent line touches at \((-2,24)\), giving \(c=2\). Question 4:
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14D - Example 4: Using differentiation to assist with graph sketching
Consider the cubic polynomial \(y=(x-1)^2(x+2)\).
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14D - Example 4: Video solution
14D - Example 4: Practice
Question 1:
Consider the function \(f(x)=\frac{x^3}{4}-\frac{3x^2}{2}+\frac{9x}{4}+4\).
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