b.) if y = 3)(⁵, find dy/dx, d²y/dx² and d³y/dx³

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Here's the solution for the differentiation problem:

(b). If $y = 3x^5$ , find $\frac{dy}{dx}$ , $\frac{d^2y}{dx^2}$ and $\frac{d^3y}{dx^3}$

Find the first derivative, $\frac{dy}{dx}$ .
Apply the power rule for differentiation, $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$y = 3x^5$ $\frac{dy}{dx} = 3 \cdot 5x^{5-1}$ $\frac{dy}{dx} = \boxed{15x^4}$
Find the second derivative, $\frac{d^2y}{dx^2}$ .
Differentiate the first derivative, $\frac{dy}{dx}$ , with respect to $x$ .
$\frac{d^2y}{dx^2} = \frac{d}{dx}(15x^4)$ $\frac{d^2y}{dx^2} = 15 \cdot 4x^{4-1}$ $\frac{d^2y}{dx^2} = \boxed{60x^3}$
Find the third derivative, $\frac{d^3y}{dx^3}$ .
Differentiate the second derivative, $\frac{d^2y}{dx^2}$ , with respect to $x$ .
$\frac{d^3y}{dx^3} = \frac{d}{dx}(60x^3)$ $\frac{d^3y}{dx^3} = 60 \cdot 3x^{3-1}$ $\frac{d^3y}{dx^3} = \boxed{180x^2}$

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