6F - Differentiation of other functions
■ The derivative of the exponential function, with base \(e\), is \[\frac{d}{dx}(e^{ax})=ae^{ax}\]
6F - Example 1: Differentiation of the exponential function
Determine the derivative of each of the following:
(a) \(f(x)=e^{-5x}\) (b) \(g(x)=-3e^{\frac{x}{2}}\) (c) \(h(x)=e^{4x-1}\) (d) \(y=\frac{1}{e^{3x}}-3e^{2x}\) |
6F - Example 1: Video solution
6F - Example 1: Practice
Question 1: For each of the following, find \(\frac{dy}{dx}\) (a) \(y=e^{-6x}\) (b) \(y=e^{4x+5}\) (c) \(y=4e^{3-5x}\) (d) \(y=\frac{2}{\sqrt{e^{6x}}}\) Question 2: If \(f(x)=-e^{5+2x}\), find \(f'(-2)\). 6F - Example 1: Solutions
Question 1: For each of the following, find \(\frac{dy}{dx}\) (a) \(\frac{dy}{dx}=-6e^{-6x}\) (b) \(\frac{dy}{dx}=4e^{4x+5}\) (c) \(\frac{dy}{dx}=-20e^{3-5x}\) (d) \(y=2e^{-3x}\therefore \frac{dy}{dx}=-6e^{-3x}\) Question 2: \(f'(x)=-2e^{5+2x}\) \(\therefore f'(-2)=-2e^{5-4}=-2e\) |
■ The derivative of the logarithmic function, with base \(e\), is \[\frac{d}{dx}(log_e(x))=\frac{1}{x}\] ■ More generally, the derivative of the natural logarithm is \[\frac{d}{dx}(log_e(g(x)))=\frac{g'(x)}{g(x)}\]
6F - Example 2: Differentiation of the natural logarithmic function
Determine the derivative of each of the following:
(a) \(f:(0,\infty)\rightarrow R, f(x)=log_e(2x)\) (b) \(g:(\frac{-3}{2},\infty)\rightarrow R, g(x)=log_e(2x+3)\) |
6F - Example 2: Video solution
6F - Example 2: Practice
Question 1: For each of the following, find \(\frac{dy}{dx}\) (a) \(y=log_e(5x)\) (b) \(y=3log_e(2-x)\) (c) \(y=log_e(3+\frac{x}{2})\) Question 2: If \(f(x)=ln(5x+6)\), find \(f'(-1)\). 6F - Example 2: Solutions
Question 1: For each of the following, find \(\frac{dy}{dx}\) (a) \(\frac{dy}{dx}=\frac{1}{x}\) (b) \(\frac{dy}{dx}=\frac{-3}{2-x}\) (c) \(\frac{dy}{dx}=\frac{1}{6+x}\) Question 2: \[f'(x)=\frac{5}{5x+6}\] \[\therefore f'(-1)=\frac{5}{5(-1)+6}=5\] |
6F - Example 3: Differentiation of the natural logarithmic function
Consider the function \[f:(-\infty,3)\rightarrow R, f(x)=2log_e(3-x)\]
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6F - Example 3: Video solution
6F - Example 3: Practice
Question 1: Find the equation of a tangent line drawn to the curve \(y=\frac{3}{2}ln(5-2x)\) at \(x=2\). 6F - Example 3: Solutions
Question 1: \[f'(x)=\frac{-3}{5-2x}\] \[\therefore f'(2)=\frac{-3}{5-2(2)}=-3\] \(f(2)=0\) \[\therefore y-0=-3(x-2)\] \[\therefore y=-3x+6\] |
There are also rules for differentiating circular functions
■ Sine function \[\frac{d}{dx}(sin(ax))=a cos(ax)\]
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■ Cosine function \[\frac{d}{dx}(cos(ax))=-a sin(ax)\]
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■ Tangent function \[\frac{d}{dx}(tan(ax))=\frac{a}{cos^2(ax)}=a sec^2(ax)\]
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6F - Example 4: Differentiation of circular functions
Determine the derivative of each of the following:
(a) \(f(x)=sin(3x)\) (b) \(g(x)=2sin(5x+7)\) (c) \(h(x)=cos(\frac{x}{4})\) (d) \(y=cos(5-2x)\) (e) \(y=tan(\frac{3x}{4})\) (f) \(y=-\frac{1}{4}tan(2-x)\) |
6F - Example 4: Video solution
6F - Example 4: Practice
Question 1: Find the derivative of each of the following (a) \(f(x)=sin(-2x)\) (b) \(g(x)=sin(\frac{1}{2}(x+\frac{\pi}{3}))\) (c) \(h(x)=-4sin(3-5x)\) Question 2: Find the derivative of each of the following (a) \(f(x)=cos(10x)\) (b) \(g(x)=5cos(3x+2)\) (c) \(h(x)=cos(\frac{\pi-x}{3})\) Question 3: Find the derivative of each of the following (a) \(f(x)=tan(\frac{x}{2})\) (b) \(g(x)=-4tan(3x)\) (c) \(h(x)=tan(7-5x)\) 6F - Example 4: Solutions
Question 1: (a) \(f'(x)=-2cos(-2x)\) (b) \(g'(x)=\frac{1}{2}cos(\frac{1}{2}(x+\frac{\pi}{3}))\) (c) \(h'(x)=20cos(3-5x)\) Question 2: Find the derivative of each of the following (a) \(f'(x)=-10sin(10x)\) (b) \(g'(x)=-15sin(3x+2)\) (c) \(h'(x)=\frac{1}{3}sin(\frac{\pi-x}{3})\) Question 3: Find the derivative of each of the following (a) \(f'(x)=\frac{1}{2{cos}^2(\frac{x}{2})}=\frac{1}{2}sec^2(\frac{x}{2})\) (b) \(g'(x)=\frac{-12}{{cos}^2(3x)}=-12sec^2(3x)\) (c) \(h'(x)=\frac{-5}{{cos}^2(7-5x)}=-5sec^2(7-5x)\) |
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Please view Topic Worksheets for any additional questions available at this point in time.
Differentiation of other functions - Worksheet A
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Past VCAA examination questions have relied on these skills when applying the chain, product and quotient rules.
Other resources from external sources may be collated here in the future.
If you know of any good resources that could be included here, please send them via email.
If you know of any good resources that could be included here, please send them via email.