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

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Here are the solutions for the remaining differentiation questions from the image.

4. Differentiate the following

The general power rule for differentiation is $\frac{d}{dx}(x^n) = nx^{n-1}$ .

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

Differentiate each term with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(7x^4) + \frac{d}{dx}(5x^3) - \frac{d}{dx}(7x)$
Apply the power rule and constant multiple rule.
$\frac{dy}{dx} = 7(4x^{4-1}) + 5(3x^{3-1}) - 7(1x^{1-1})$ $\frac{dy}{dx} = 7(4x^3) + 5(3x^2) - 7(1x^0)$ $\frac{dy}{dx} = 28x^3 + 15x^2 - 7(1)$ $\frac{dy}{dx} = 28x^3 + 15x^2 - 7$
The final answer for 4(1) is $\boxed{28x^3 + 15x^2 - 7}$ .

(2) $y = x^3$
Differentiate $y$ with respect to $x$ using the power rule.
$\frac{dy}{dx} = 3x^{3-1}$ $\frac{dy}{dx} = 3x^2$
The final answer for 4(2) is $\boxed{3x^2}$ .

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

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

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