Find The Derivative Or The Integral Of The Given Function:, 1. Y= -7x2074

Find the derivative or the integral of the given function:

1. y= -7x⁴ - 3³√x
2. y= ³√9x²+10x-1

1. $y = -7x^{4} - 3\sqrt3{x}$

Derivative:

$\frac{dy}{dx} = -7x^{4} - 3\sqrt3{x}$

$-28x^{3} - 3\frac{dy}{dx}(x)^{\frac{1}{3}}$

$-28x^{3} - 3(\frac{1}{3})(x)^{\frac{-2}{3}}$

$-28x^{3} - \frac{1}{\sqrt3{x^{2}}}$

Antiderivative:

∫ $(-7x^{4} - 3\sqrt3{x}) dx$

$\frac{-7}{5}x^{5} - 3$ ∫ $(x)^{\frac{1}{3}}$

$\frac{-7}{5}x^{5} - 3 (\frac{3}{4})(x)^{\frac{4}{3}}$

$\frac{-7}{5}x^{5} - \frac{9}{4}x^{\frac{4}{3}} + C$

2. $y = \sqrt3{9x^{2}} + 10x - 1$

Derivative:

$\frac{dy}{dx} = \sqrt3{9x^{2}} + 10x - 1$

$(9x^{2})^{\frac{1}{3}} + 10x - 1$

$\frac{1}{3}(9x^{2})^{\frac{-2}{3}} + 10$

$\frac{1}{3\sqrt3{(9x^{2})^{2}}} + 10$

(Note: I am not sure on this part.)

Antiderivative:

∫ $(\sqrt3{9x^{2}} + 10x - 1) dx$

$(9x^{2})^{\frac{1}{3}} + 10x - 1$

$(\frac{1}{18x})(\frac{3}{4})(9x^{2})^{\frac{4}{3}} + 5x^{2} - x + C$

$\frac{1}{24x}(9x^{2})^{\frac{4}{3}} + 5x^{2} - x + C$

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