Let 
(a) Use the definition of the derivative to compute 
(b) Find the equation of the tangent line to 
at 
| Background Information:
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| Recall
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Solution:
(a)
| Step 1:
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| Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=3{\sqrt {2x+5}}.}
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| Using the limit definition of the derivative, we have
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Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3{\sqrt {2(x+h)+5}}-3{\sqrt {2x+5}}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3{\sqrt {2x+2h+5}}-3{\sqrt {2x+5}}}{h}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {{\sqrt {2x+2h+5}}-{\sqrt {2x+5}}}{h}}.}\end{array}}}
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| Step 2:
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| Now, we multiply the numerator and denominator by the conjugate of the numerator.
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| Hence, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {({\sqrt {2x+2h+5}}-{\sqrt {2x+5}})}{h}}{\frac {({\sqrt {2x+2h+5}}+{\sqrt {2x+5}})}{({\sqrt {2x+2h+5}}+{\sqrt {2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {(2x+2h+5)-(2x+5)}{h({\sqrt {2x+2h+5}}+{\sqrt {2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {2h}{h({\sqrt {2x+2h+5}}+{\sqrt {2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {2}{{\sqrt {2x+2h+5}}+{\sqrt {2x+5}}}}}\\&&\\&=&\displaystyle {3{\frac {2}{{\sqrt {2x+5}}+{\sqrt {2x+5}}}}}\\&&\\&=&\displaystyle {{\frac {3}{\sqrt {2x+5}}}.}\end{array}}}
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(b)
| Step 1:
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| We start by finding the slope of the tangent line to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=3{\sqrt {2x+5}}}
at
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| Using the derivative calculated in part (a), the slope is
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {m}&=&\displaystyle {f'(2)}\\&&\\&=&\displaystyle {\frac {3}{\sqrt {2(2)+5}}}\\&&\\&=&\displaystyle {\frac {3}{\sqrt {9}}}\\&&\\&=&\displaystyle {1.}\end{array}}}
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| Step 2:
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Now, the tangent line to  at 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 (2,9)}
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| has slope 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 m=1}
and passes through the point 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 (2,9).}
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| Hence, the equation of this line is
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- 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 y=(x-2)+9.}
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| If we simplify, we get
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- 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 y=x+7.}
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| 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 \frac{3}{\sqrt{2x+5}}}
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| (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 y=x+7}
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