Difference between revisions of "009A Sample Final A, Problem 2"
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|These are problems involving some more advanced rules of differentiation. In particular, they use | |These are problems involving some more advanced rules of differentiation. In particular, they use | ||
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| − | |'''The Chain Rule:''' If | + | |'''The Chain Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -20%;">g</math> are differentiable functions, then |
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|<br> <math>(f\circ g)'(x) = f'(g(x))\cdot g'(x).</math> | |<br> <math>(f\circ g)'(x) = f'(g(x))\cdot g'(x).</math> | ||
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| − | |<br>'''The Product Rule:''' If | + | |<br>'''The Product Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -20%;">g</math> are differentiable functions, then |
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|<br> <math>(fg)'(x) = f'(x)\cdot g(x)+f(x)\cdot g'(x).</math> | |<br> <math>(fg)'(x) = f'(x)\cdot g(x)+f(x)\cdot g'(x).</math> | ||
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| − | |<br>'''The Quotient Rule:''' If | + | |<br>'''The Quotient Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -20%;">g</math> are differentiable functions and <math style="vertical-align: -25%;">g(x) \neq 0</math>, then |
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|<br> <math>\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. </math> | |<br> <math>\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. </math> | ||
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!Part (b): | !Part (b): | ||
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| − | |Both parts (b) and (c) attempt to confuse you by including the familiar constants | + | |Both parts (b) and (c) attempt to confuse you by including the familiar constants <math style="vertical-align: 0%;">e</math> and <math style="vertical-align: 0%;">\pi</math>. Remember - they are just constants, like 10 or 1/2. With that in mind, we really just need to apply the chain rule to find |
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| <math>g'(x)=0-2\sin\left(\sqrt{x-2}\right)\cdot\frac{1}{2}\cdot(x-2)^{-1/2}\cdot1=\,-\frac{\sin\left(\sqrt{x-2}\right)}{\sqrt{x-2}}.</math> | | <math>g'(x)=0-2\sin\left(\sqrt{x-2}\right)\cdot\frac{1}{2}\cdot(x-2)^{-1/2}\cdot1=\,-\frac{\sin\left(\sqrt{x-2}\right)}{\sqrt{x-2}}.</math> | ||
Revision as of 21:34, 26 March 2015
2. 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)=\frac{3x^{2}-5}{x^{3}-9}.}
(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)=\pi+2\cos(\sqrt{x-2}).}
(c) 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 h(x)=4x\sin(x)+e(x^{2}+2)^{2}.}
| Foundations: |
|---|
| These are problems involving some more advanced rules of differentiation. In particular, they use |
| The Chain Rule: If 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} and 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} are differentiable functions, then |
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\circ g)'(x) = f'(g(x))\cdot g'(x).} |
The Product Rule: If 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} and 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} are differentiable functions, then |
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 (fg)'(x) = f'(x)\cdot g(x)+f(x)\cdot g'(x).} |
The Quotient Rule: If 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} and 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} are differentiable functions and 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) \neq 0} , then |
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 \left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. } |
Solution:
| Part (a): |
|---|
| We need to use the 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 f'(x) = \frac {\left(3x^{2}-5\right)' \cdot \left(x^{3}-9 \right)- \left( 3x^{2}-5 \right) \cdot \left( x^{3}-9\right)'}{\left(x^{3}-9\right)^2} } |
| 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{6x^{4}-54x-9x^{4}+15x^{2}}{(x^{3}-9)^{2}}} |
| 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{-3x^{4}+15x^{2}-54x}{(x^{3}-9)^{2}}.} |
| Part (b): |
|---|
| Both parts (b) and (c) attempt to confuse you by including the familiar constants 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 e} and 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 \pi} . Remember - they are just constants, like 10 or 1/2. With that in mind, we really just need to apply the chain rule to find |
| 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)=0-2\sin\left(\sqrt{x-2}\right)\cdot\frac{1}{2}\cdot(x-2)^{-1/2}\cdot1=\,-\frac{\sin\left(\sqrt{x-2}\right)}{\sqrt{x-2}}.} |
| Part (c): |
|---|
| We can choose to expand the second term, finding |
| 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 e(x^{2}+2)^{2}=ex^{4}+4ex^{2}+4e.} |
| We then only require the product rule on the first term, so |
| 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 h'(x)=(4x)'\cdot\sin(x)+4x\cdot(\sin(x))'+\left(ex^{4}+4ex^{2}+4e\right)'=4\sin(x)+4x\cos(x)+4ex^{3}+8ex.} |