Question

$\bold \red{2x^3-3x^2-36x+2}$


❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ
❖ ɴᴏ ꜱᴘᴀᴍᴍɪɴɢ
❖ᴏɴʟʏ ꜰᴏʀ ᴍᴏᴅᴇʀᴀᴛᴏʀꜱ, ʙʀᴀɪɴʟʏ ꜱᴛᴀʀꜱ ᴀɴᴅ ᴏᴛʜᴇʀ ʙᴇꜱᴛ ᴜꜱᴇʀꜱ​​​​​​​​​​​​​​​​​​​​​​​​

Answer 1

SOLUTION

• Differentiate separately both sides of the equation (treat y as a function of x):

$\sf \bold \red{\frac{d}{dx} \left(y{\left(x \right)}\right)}$

$\bold \red{= \frac{d}{dx} \left(2 x^{3} - 3 x^{2} - 36 x + 2\right)}$

Differentiate the RHS of the equation.

The derivative of a sum/difference is the sum/difference of derivatives:

$\small \bold{\color{r}{\left(\frac{d}{dx} \left(2 x^{3} - 3 x^{2} - 36 x + 2\right)\right)} = \color{r}{\left(\frac{d}{dx} \left(2 x^{3}\right) - \frac{d}{dx} \left(3 x^{2}\right) - \frac{d}{dx} \left(36 x\right) + \frac{d}{dx} \left(2\right)\right)}}$

The derivative of a constant is 0:

$\small \bold{\color{y}{\left(\frac{d}{dx} \left(2\right)\right)} - \frac{d}{dx} \left(36 x\right) - \frac{d}{dx} \left(3 x^{2}\right) + \frac{d}{dx} \left(2 x^{3}\right) = \color{red}{\left(0\right)} - \frac{d}{dx} \left(36 x\right) - \frac{d}{dx} \left(3 x^{2}\right) + \frac{d}{dx} \left(2 x^{3}\right)}$

Apply the constant multiple rule

$\small\bold {{\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) \: with \: c = 2 \: and \: f{\left(x \right)} = x^{3}}}$

$\begin{gathered}\color{red} \sf{\left(\frac{d}{dx} \left(2 x^{3}\right)\right)} - \frac{d}{dx} \left(36 x\right) - \frac{d}{dx} \left(3 x^{2}\right) = \color{red}{\left(2 \frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(36 x\right) - \frac{d}{dx} \left(3 x^{2}\right) \\ \sf \: Apply \: the \: constant \: multiple \: rule \\ \: \sf\frac{d}{dx} \left(c f{\left(x\right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) \: with \: c = 3 \: and \: f{\left(x \right)} = x^{2}\end{gathered}$

$\bold{- \color{black}{\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} - \frac{d}{dx} \left(36 x\right) + 2 \frac{d}{dx} \left(x^{3}\right) = - \color{green}{\left(3 \frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(36 x\right) + 2 \frac{d}{dx} \left(x^{3}\right)}$

Apply the power rule $

$\bold{\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} \: with \: n = 2}$

withn=2

$\bold{- 3 \color{red}{\left(\frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(36 x\right) + 2 \frac{d}{dx} \left(x^{3}\right) = - 3 \color{red}{\left(2 x\right)} - \frac{d}{dx} \left(36 x\right) + 2 \frac{d}{dx} \left(x^{3}\right)}$

Apply the power rule

$\bold{\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} \: with \: n = 3}$

$\bold{- 6 x + 2 \color{red}{\left(\frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(36 x\right) = - 6 x + 2 \color{red}{\left(3 x^{2}\right)} - \frac{d}{dx} \left(36 x\right)}$

Apply the constant multiple rule

$\bold{\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) \: with \: c = 36 \: and \: f{\left(x \right)} = x:}$

$\bold{6 x^{2} - 6 x - \color{green}{\left(\frac{d}{dx} \left(36 x\right)\right)} = 6 x^{2} - 6 x - \color{red}{\left(36 \frac{d}{dx} \left(x\right)\right)}}$

Apply the power rule

$\bold{\bold \pink{\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} \: with \: n = 1, in \: other \: words, \frac{d}{dx} \left(x\right) = 1}}$

$\small \bold{6 x^{2} - 6 x - 36 {} \: {\left(\frac{d}{dx} \left(x\right)\right)} = 6 x^{2} - 6 x - 36 \color{red}{\left(1\right)}}$

Simplify:

$\bold{6 x^{2} - 6 x - 36 = 6 \left(x - 3\right) \left(x + 2\right)}$

$\sf{ \small \bold {Thus, \frac{d}{dx} \left(2 x^{3} - 3 x^{2} - 36 x + 2\right) = 6 \left(x - 3\right) \left(x + 2\right).}}$

$\therefore\small{ \bold \red{ \frac{dy}{dx} = 6 \left(x - 3\right) \left(x + 2\right)}}$

ANSWER

$\sf\frac{dy}{dx} = 6 \left(x - 3\right) \left(x + 2\right)$

6(x−3)(x+2)