Differentiate the following: (1) y = 7x4 + 5x3 - 7x (2) y = x3 (3) y = 8x2 - x + 5x (4) y = xⁿ

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4. Differentiate the following

(1) $y = 7x^4 + 5x^3 - 7x$

Differentiate each term using the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$\frac{dy}{dx} = \frac{d}{dx}(7x^4) + \frac{d}{dx}(5x^3) - \frac{d}{dx}(7x)$
Apply the power rule to each term.
$\frac{d}{dx}(7x^4) = 7 \times 4x^{4-1} = 28x^3$ $\frac{d}{dx}(5x^3) = 5 \times 3x^{3-1} = 15x^2$ $\frac{d}{dx}(7x) = 7 \times 1x^{1-1} = 7x^0 = 7$
Combine the results.
$\frac{dy}{dx} = 28x^3 + 15x^2 - 7$
The final answer for (1) is $\boxed{28x^3 + 15x^2 - 7}$ .

(2) $y = x^3$
Differentiate using the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$\frac{dy}{dx} = 3x^{3-1}$ $\frac{dy}{dx} = 3x^2$
The final answer for (2) is $\boxed{3x^2}$ .

(3) $y = 8x^2 - x + 5x$
Simplify the expression for $y$ first.
$y = 8x^2 + (-1 + 5)x$ $y = 8x^2 + 4x$
Differentiate each term using the power rule.
$\frac{dy}{dx} = \frac{d}{dx}(8x^2) + \frac{d}{dx}(4x)$
Apply the power rule to each term.
$\frac{d}{dx}(8x^2) = 8 \times 2x^{2-1} = 16x^1 = 16x$ $\frac{d}{dx}(4x) = 4 \times 1x^{1-1} = 4x^0 = 4$
Combine the results.
$\frac{dy}{dx} = 16x + 4$
The final answer for (3) is $\boxed{16x + 4}$ .

(4) $y = x^n$
Differentiate using the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$\frac{dy}{dx} = nx^{n-1}$
The final answer for (4) is $\boxed{nx^{n-1}}$ .

Differentiate the following: (1) y = 7x4 + 5x3 − 7x (2) y = x3 (3) y = 8x2 − x + 5x (4) y = xⁿ