Derivative Calculation

Question

I've some equation about "Derivatives" to ask about. Please, show me how to do that step by step: $$f(x) = \frac{3x^2+1}{2}.$$ $f'(x)= ?$

Answer 1

$\frac{1}{2}\frac{d}{dx}(3x^2 + 1) = \frac{1}{2}(\frac{d}{dx}3x^2 + \frac{d}{dx}1) = \frac{1}{2}(3\frac{d}{dx}x^2 + \frac{d}{dx}1)$

Since 1 is a constant its derivative becomes 0 and as for $x^2$ we have a rule that states that if $f(x) = x^r$ then $f'(x) = r\cdot x^{r-1}$. With that in mind we get

$\frac{1}{2}(3\frac{d}{dx}x^2 + \frac{d}{dx}1) = \frac{1}{2}(3(2x) + 0) = \frac{1}{2}6x = 3x$

Answer 2

Each step should follow one of the derivative rules that you know about. The notation "$\frac{d}{dx}$" in what follows means "take the derivative of".

$$ \begin{align*} f'(x) &= \frac{d}{dx}\left[ \frac{3x^2 + 1}{2} \right]\\ &= \frac{d}{dx}\left[ \frac{1}{2}(3x^2 + 1) \right], \quad\textrm{(algebra)}\\ &= \frac{1}{2}\frac{d}{dx}\left[3x^2 + 1 \right], \quad\textrm{(constant multiple rule)}\\ &= \frac{1}{2}\left( \frac{d}{dx}[3x^2] + \frac{d}{dx}[1] \right), \quad \textrm{(sum/difference rule)}\\ &= \frac{1}{2}\left( 3\frac{d}{dx}[x^2] + \frac{d}{dx}[1] \right), \quad \textrm{(constant mult. rule again)}\\ &= \frac{1}{2}\left( 3(2x) + \frac{d}{dx}[1] \right), \quad \textrm{(power rule)}\\ &= \frac{1}{2}\left( 3(2x) + 0 \right), \quad \textrm{(derivative of a constant is 0 -- really just power rule)}\\ &= 3x, \quad \textrm{(algebra to simplify answer)} \end{align*} $$ Now as you do more and more of these problems, you'll find which steps you can do in your head, until you get to the point where it becomes a one-line problem!

Hope this helps!

Answer 3

It's simple: Just apply the definition of the derivative ($f$ is a polynomial so is differentiable, which we can prove).

$f'(x) = \lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$

$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim \limits_{h \to 0} \frac{(3(x+h)^2 + 1) -(3x^2 +1)}{2h} = $ . . . ?

All it takes is a little manipulation. You should find some very important things will cancel out and the limit will be easy to take.

Answer 4

$f(x) = \frac32 x^2 + \frac12$. Just use the Power Rule.

Answer 5

Type your equation into wolfram alpha, put d/dx in front of it, click on the '=' on the right of the text entry field, wait for it to finish and finally click on 'show steps' in the top right corner of the results-box.

Alternatively click on this link, and click on the 'show steps' in the top right corner of the results-box.