Question

If y = 2x³ - 3x² - 36x + 2 . Find dy/dx !

No Irrelevant Stuff Plz :) ​

Answer 1

SOLUTION

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

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

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

Differentiate the RHS of the equation.

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

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

The derivative of a constant is 0:

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

Apply the constant multiple rule

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

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

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

Apply the power rule $$$

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

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

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

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

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

Apply the power rule

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

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

Simplify:

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

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

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

ANSWER

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

Answer 2

${\boxed{\boxed{\begin{array}{cc}\bf \: \to \:given : \\ \\ \rm \: y = 2 {x}^{3} - 3 {x}^{2} - 36x + 2 \\ \\ \sf \: we \: have \: to \: differentiate \: it \: w.r.t. \: \: \bf \: x \\ \\ \therefore \: \rm \: \frac{dy}{dx} = \frac{d}{dx} (2 {x}^{3} - 3 {x}^{2} - 36x + 2) \\ \\ \pink{{\boxed{\begin{array}{cc}\bf \:we \: know \: that : \\ \bf \: \frac{d}{dx}(u \pm \: v) = \frac{d}{dx} \: u \pm \frac{d}{dx} \: v\end{array}}}} \\ \orange{ \sf \: apply \: this} \\ \\ \rm = \frac{d}{dx}2 {x}^{3} - \frac{d}{dx} 3 {x}^{2} - \frac{d}{dx} 36x + \frac{d}{dx} \: 2 \\ \\ \pink{{\boxed{\begin{array}{cc}\bf \: we \: know \: that : \\ \bf \: \frac{d}{dx} \: [constant(c)] = 0 \\ \\ \bf \: \frac{d}{dx} \: c {x}^{n} = cn {x}^{n - 1} \end{array}}}} \\ \orange{ \sf \: apply \: this} \\ \\ \rm = 2 \times 3 \times {x}^{3 - 1} - 3 \times 2 \times {x}^{2 - 1} - 36 \times {x}^{1 - 1} + 0 \\ \\ \rm = 6 {x}^{2} - 6x - 36 \\ \\ \\ \\ \blue{ \boxed{\therefore \rm \: \frac{dy}{dx} = 6 {x}^{2} - 6x - 36}}\end{array}}}}$