009A Sample Final 1, Problem 3 Detailed Solution
Find the derivatives of the following functions.
(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln \bigg(\frac{x^2-1}{x^2+1}\bigg)}
(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=3\sin (4x)+4\tan (\sqrt{1+x^3})}
| Background Information: |
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| 1. Chain Rule |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)} |
| 2. Quotient Rule |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}} |
| 3. Trig Derivatives |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}(\sin x)=\cos x,\quad\frac{d}{dx}(\tan x)=\sec^2 x} |
Solution:
(a)
| Step 1: |
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| Using the Chain Rule, we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{f'(x)} & = & \displaystyle{\frac{1}{\bigg(\frac{x^2-1}{x^2+1}\bigg)}\cdot\frac{d}{dx}\bigg(\frac{x^2-1}{x^2+1}\bigg)}\\ &&\\ & = & \displaystyle{\frac{x^2+1}{x^2-1}\cdot\frac{d}{dx}\bigg(\frac{x^2-1}{x^2+1}\bigg).}\\ \end{array}} |
| Step 2: |
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| Now, we need to calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\bigg(\frac{x^2-1}{x^2+1}\bigg).} |
| To do this, we use the Quotient Rule. So, we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{f'(x)} & = & \displaystyle{\frac{x^2+1}{x^2-1}\cdot\frac{d}{dx}\bigg(\frac{x^2-1}{x^2+1}\bigg)}\\ &&\\ & = & \displaystyle{\frac{x^2+1}{x^2-1}\cdot\frac{(x^2+1)(2x)-(x^2-1)(2x)}{(x^2+1)^2}}\\ &&\\ & = & \displaystyle{\frac{x^2+1}{x^2-1}\cdot\frac{4x}{(x^2+1)^2}\cdot}\\ &&\\ & = & \displaystyle{\frac{4x}{(x^2-1)(x^2+1)}}\\ &&\\ & = & \displaystyle{\frac{4x}{x^4-1}.}\\ \end{array}} |
(b)
| Step 1: |
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| We need to use the Chain Rule. We have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{g'(x)} & = & \displaystyle{3\cos(4x)\cdot (4x)'+4\sec^2(\sqrt{1+x^3})\cdot \frac{d}{dx}(\sqrt{1+x^3})}\\ &&\\ & = & \displaystyle{3\cos(4x)\cdot 4+4\sec^2(\sqrt{1+x^3})\cdot \frac{d}{dx}(\sqrt{1+x^3})}\\ &&\\ & = & \displaystyle{12\cos(4x)+4\sec^2(\sqrt{1+x^3})\cdot \frac{d}{dx}(\sqrt{1+x^3})}\\ \end{array}} |
| Step 2: |
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| We need to calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}(\sqrt{1+x^3}).} |
| We use the Chain Rule again to get |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{g'(x)} & = & \displaystyle{12\cos(4x)+4\sec^2(\sqrt{1+x^3})\cdot\frac{d}{dx}\sqrt{1+x^3}}\\ &&\\ & = & \displaystyle{12\cos(4x)+4\sec^2(\sqrt{1+x^3})\frac{1}{2}(1+x^3)^{-\frac{1}{2}}\cdot \frac{d}{dx}(1+x^3)}\\ &&\\ & = & \displaystyle{12\cos(4x)+4\sec^2(\sqrt{1+x^3})\frac{1}{2}(1+x^3)^{-\frac{1}{2}} (3x^2)}\\ &&\\ & = & \displaystyle{12\cos(4x)+\frac{6\sec^2(\sqrt{1+x^3})x^2}{\sqrt{1+x^3}}.}\\ \end{array}} |
| Final Answer: |
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| (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{4x}{x^4-1}} |
| (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g'(x)=12\cos(4x)+\frac{6\sec^2(\sqrt{1+x^3})x^2}{\sqrt{1+x^3}}} |