Question 1
Differentiate the following functions:
(a)\(f(x) = 3x^3 - 5x + 1\)
(b)\(f(x) = \sqrt[4]{x}\)
(c)\(f(x) = \dfrac{1}{x^2}\)
(d)\(f(x) = 5x^5 - 3 + \dfrac{4}{x^4}\)
(e)\(f(x) = \dfrac{1}{\sqrt{x}}\)
(f)\(f(x) = \sqrt[5]{x^3}\)
Answer
\(f'(x) = 9x^2 - 5\)
Rewrite
\(f(x) = x^{1/4}\)
Answer
\(f'(x) = \dfrac{1}{4}x^{-3/4}\)
Answer
\(f'(x) = -2x^{-3} = -\dfrac{2}{x^3}\)
Rewrite
\(f(x) = 5x^5 - 3 + 4x^{-4}\)
Answer
\(f'(x) = 25x^4 - 16x^{-5}\)
Rewrite
\(f(x) = x^{-1/2}\)
Answer
\(f'(x) = -\dfrac{1}{2}x^{-3/2}\)
Rewrite
\(f(x) = x^{3/5}\)
Answer
\(f'(x) = \dfrac{3}{5}x^{-2/5}\)
Question 2
By first simplifying, differentiate the following functions:
(a)\(f(x) = (x+3)(3x-1)\)
(b)\(f(x) = 3x^2\sqrt{x}\)
(c)\(f(x) = \dfrac{4x^3 - 6x}{2x}\)
(d)\(f(x) = \!\left(\dfrac{3x^2}{x}\right)^{\!3}\)
(e)\(f(x) = \dfrac{2}{\sqrt[3]{x}}\)
Expand
\(f(x) = 3x^2 - x + 9x - 3 = 3x^2 + 8x - 3\)
Rewrite
\(f(x) = 3x^2 \cdot x^{1/2} = 3x^{5/2}\)
Answer
\(f'(x) = \dfrac{15}{2}x^{3/2}\)
Simplify
\(f(x) = 2x^2 - 3\)
Simplify
\(\dfrac{3x^2}{x} = 3x\), so \(f(x) = (3x)^3 = 27x^3\)
Rewrite
\(f(x) = 2x^{-1/3}\)
Answer
\(f'(x) = -\dfrac{2}{3}x^{-4/3}\)
Question 3
Expand \((2x-3)^4\) using the binomial expansion. Hence, if \(f(x) = (2x-3)^4\), find \(f'(x)\).
Expand
\((2x-3)^4 = 16x^4 - 4\cdot8x^3\cdot3 + 6\cdot4x^2\cdot9 - 4\cdot2x\cdot27 + 81\)
\(= 16x^4 - 96x^3 + 216x^2 - 216x + 81\)
Answer
\(f'(x) = 64x^3 - 288x^2 + 432x - 216\)
Question 4
If \(A = \pi r^2 + \dfrac{6\pi}{r}\), find \(\dfrac{dA}{dr}\).
Rewrite
\(A = \pi r^2 + 6\pi r^{-1}\)
Answer
\(\dfrac{dA}{dr} = 2\pi r - \dfrac{6\pi}{r^2}\)
Question 5
If \(P = \dfrac{4t^3}{3} - 5t^2 + 12\), find \(\dfrac{dP}{dt}\).
Answer
\(\dfrac{dP}{dt} = 4t^2 - 10t\)
Question 6
\(f(x) = \dfrac{x^3}{3} - 2x^2 + 9x - 4\). Find \(f'(2)\).
Differentiate
\(f'(x) = x^2 - 4x + 9\)
Substitute \(x=2\)
\(f'(2) = 4 - 8 + 9\)
Question 7
Find the gradient of the following functions at the given \(x\) values.
(a)\(y = -x^3 + 2\sqrt{x} + 5\) at \(x = 4\)
(b)\(f(x) = 3x^5 - 2x^2 + x\) at \(x = -1\)
Differentiate
\(y' = -3x^2 + x^{-1/2}\)
Substitute \(x=4\)
\(y'(4) = -48 + \frac{1}{2}\)
Differentiate
\(f'(x) = 15x^4 - 4x + 1\)
Substitute \(x=-1\)
\(f'(-1) = 15 + 4 + 1\)
Question 8
\(f(x) = x^2 - 4x + 7\). Find:
(a)\(f'(5)\)
(b)the value of \(x\) where \(f'(x) = 0\)
(c)the values of \(x\) where \(f'(x) = 4\)
Differentiate
\(f'(x) = 2x - 4\)
Answer
\(f'(5) = 10 - 4 = 6\)
Solve
\(2x-4=0 \Rightarrow x=2\)
Solve
\(2x-4=4 \Rightarrow 2x=8\)
Question 9
\(f(x) = \dfrac{3}{x^2} + 4x\). Solve \(f'(x) = 1\).
Rewrite & differentiate
\(f(x) = 3x^{-2} + 4x \Rightarrow f'(x) = -6x^{-3} + 4\)
Solve
\(-\dfrac{6}{x^3} + 4 = 1 \Rightarrow \dfrac{6}{x^3} = 3 \Rightarrow x^3 = 2\)
Answer
\(x = \sqrt[3]{2}\)
Question 10
\(f(x) = \dfrac{k}{x^3}\). Find \(f'(2)\) in terms of \(k\).
Rewrite & differentiate
\(f(x) = kx^{-3} \Rightarrow f'(x) = -3kx^{-4}\)
Answer
\(f'(2) = -\dfrac{3k}{16}\)
Question 11
\(f(x) = \dfrac{x^3}{3} - \dfrac{x^2}{2} - 2x\). Find:
(a)\(f'(2)\)
(b)the values of \(x\) where \(f'(x) = 0\)
(c)the values of \(x\) where \(f'(x) = 6\)
Differentiate
\(f'(x) = x^2 - x - 2\)
Answer
\(f'(2) = 4 - 2 - 2 = 0\)
Solve
\(x^2 - x - 2 = 0 \Rightarrow (x-2)(x+1) = 0\)
Answer
\(x = 2\) or \(x = -1\)
Solve
\(x^2 - x - 2 = 6 \Rightarrow x^2 - x - 8 = 0 \Rightarrow x = \dfrac{1 \pm \sqrt{33}}{2}\)
Answer
\(x = \dfrac{1 + \sqrt{33}}{2}\) or \(x = \dfrac{1 - \sqrt{33}}{2}\)