"How do you find the slope of a..." was answered by our community

painter555ui8n

Answered question

2022-04-10

How do you find the slope of a tangent line to the graph of the function

f (x) = \frac{x^{2}}{x + 1}

at (2, 4/3)?

pobijedi6wro

Beginner2022-04-11Added 15 answers

Explanation:
Reminder
The slope of the tangent at x = a on f(x) is f'(a)
To differentiate f(x) use the quotient rule
given

f (x) = \frac{g (x)}{h (x)}

then

f^{'} (x) = \frac{h (x) g^{'} (x) - g (x) h^{'} (x)}{{(h (x))}^{2}}

here

g (x) = x^{2} \Rightarrow g^{'} (x) = 2 x

and

h (x) = x + 1 \Rightarrow h^{'} (x) = 1

\Rightarrow f^{'} (x) = \frac{(x + 1) \cdot 2 x - x^{2} \cdot 1}{{(x + 1)}^{2}} = \frac{2 x^{2} + 2 x - x^{2}}{{(x + 1)}^{2}}

= \frac{x^{2} + 2 x}{{(x + 1)}^{2}}

\Rightarrow f^{'} (2) = \frac{4 + 4}{3^{2}} = \frac{8}{9}

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