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hi I don’t know how to answer part B of the Question, I’m in high school calculus one, and this is a homework

hi I don’t know how to answer part B of the Question, I’m in high school calculus-example-1
User Gamadril
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1 Answer

4 votes

SOLUTION

The given function is


f(x)=7x-3x^2

Using the limit definition


f^(\prime)(x)=\lim _(h\to0)(f(x+h)-f(x))/(h)

Substitute x+h for x

This gives


f^(\prime)(x)=\lim _(h\to0)(7(x+h)+3(x+h)^2-(7x-3x^2))/(h)

Simplify the limit


\begin{gathered} f^(\prime)(x)=\lim _(h\to0)\frac{7x+7h-3(x^2+2hx+h^2)^{}-7x+3x^2}{h} \\ \end{gathered}

This further gives


f^(\prime)(x)=\lim _(h\to0)(7x+7h-3x^2-6hx-3h^2-7x+3x^2)/(h)

Simplify further


f^(\prime)(x)=\lim _(h\to0)(7h-6hx-3h^2)/(h)

Simplify the fraction


\begin{gathered} f^(\prime)(x)=\lim _(h\to0)((7h)/(h)-(6hx)/(h)-(3h^2)/(h)) \\ f^(\prime)(x)=\lim _(h\to0)(7-6x-3h) \end{gathered}

Find the limit


\begin{gathered} f^(\prime)(x)=(7-6x-3(0)) \\ f^(\prime)(x)=7-6x \end{gathered}

Therefore, the solution is


f^(\prime)(x)=7-6x

User DanW
by
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