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Use the definition of the derivative to find f'(x).
f(x)=[tex]\sqrt{2x}[/tex]
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3.2 Question 59
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Re: 3.2 Question 59
by Eigenvalue » Thu Sep 24, 2026 12:16 am
f'(x)=[tex]\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
=[tex]\lim_{h \to 0} \frac{ \sqrt{2x+2h}- \sqrt{2x} }{h}[/tex]
Multiple both sides by [tex]\sqrt{2x+2h}[/tex]-[tex]\sqrt{2x}[/tex]
=[tex]\lim_{h\to 0} \frac{2h}{h( \sqrt{2x+2h}- \sqrt{2x} }[/tex]
Evaluate the limits:
=[tex]\frac{2}{2 \sqrt{2x} }[/tex]
=[tex]\frac{1}{ \sqrt{2x} }[/tex]
=[tex]\frac{ \sqrt{2x} }{2x}[/tex]
=[tex]\lim_{h \to 0} \frac{ \sqrt{2x+2h}- \sqrt{2x} }{h}[/tex]
Multiple both sides by [tex]\sqrt{2x+2h}[/tex]-[tex]\sqrt{2x}[/tex]
=[tex]\lim_{h\to 0} \frac{2h}{h( \sqrt{2x+2h}- \sqrt{2x} }[/tex]
Evaluate the limits:
=[tex]\frac{2}{2 \sqrt{2x} }[/tex]
=[tex]\frac{1}{ \sqrt{2x} }[/tex]
=[tex]\frac{ \sqrt{2x} }{2x}[/tex]
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Re: 3.2 Question 59
by nycmath » Thu Sep 24, 2026 4:17 am
Eigenvalue wrote:f'(x)=[tex]\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
=[tex]\lim_{h \to 0} \frac{ \sqrt{2x+2h}- \sqrt{2x} }{h}[/tex]
Multiple both sides by [tex]\sqrt{2x+2h}[/tex]-[tex]\sqrt{2x}[/tex]
=[tex]\lim_{h\to 0} \frac{2h}{h( \sqrt{2x+2h}- \sqrt{2x} }[/tex]
Evaluate the limits:
=[tex]\frac{2}{2 \sqrt{2x} }[/tex]
=[tex]\frac{1}{ \sqrt{2x} }[/tex]
=[tex]\frac{ \sqrt{2x} }{2x}[/tex]
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