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Quick differentiation question

Hi, I was wondering how to go about differentiating a term such as

\frac{x^2}{2}

?
Thanks.

Reply 1

Aaliyah.97

You use the indices rule to change the fraction then you differentiate it

Wait sorry ignore that

(edited 8 years ago)

Reply 2

Zacken

Original post by Zenzation

Hi, I was wondering how to go about differentiating a term such as

\frac{x^2}{2}

?
Thanks.

Differentiation is a linear operator so:

Unparseable latex formula:

\displaystyle[br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \left(af(x)\right) = a\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \left(f(x)\right)\end{equation*}

And:

Unparseable latex formula:

\displaystyle[br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \left( f(x) + g(x)\right) = \displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \left( f(x)\right) + \frac{\mathrm{d}}{\mathrm{d}x} \left(g(x)\right)\end{equation*}

Secondly, we have the rule that:

Unparseable latex formula:

\displaystyle [br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \left(x^{n}\right) = nx^{n-1}\end{equation*}

Can you combine the above to differentiate your term?

Hint: here, you'll want to use the first rule with

a = \frac{1}{2}

and the third rule with

n = 2

(edited 8 years ago)

Reply 3

Zenzation

Original post by Zacken

Differentiation is a linear operator so:

Unparseable latex formula:

\displaystyle[br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \left(af(x)\right) = a\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \left(f(x)\right)\end{equation*}

And:

Unparseable latex formula:

Secondly, we have the rule that:

Unparseable latex formula:

\displaystyle [br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \left(x^{n}\right) = nx^{n-1}\end{equation*}