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

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

4. Differentiate the following

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

(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 to each term.
$\frac{dy}{dx} = 7(4x^{4-1}) + 5(3x^{3-1}) - 7(1x^{1-1})$ $\frac{dy}{dx} = 28x^3 + 15x^2 - 7x^0$ $\frac{dy}{dx} = 28x^3 + 15x^2 - 7$
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) $y = 8x^3 - x + 5x$
Simplify the expression for $y$ by combining like terms.
$y = 8x^3 + (-x + 5x)$ $y = 8x^3 + 4x$
Differentiate each term with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(8x^3) + \frac{d}{dx}(4x)$
Apply the power rule to each term.
$\frac{dy}{dx} = 8(3x^{3-1}) + 4(1x^{1-1})$ $\frac{dy}{dx} = 24x^2 + 4x^0$ $\frac{dy}{dx} = 24x^2 + 4$
The final answer for 4(3) is $\boxed{24x^2 + 4}$ .

(4) $y = x^{-7}$
Differentiate $y$ with respect to $x$ using the power rule.
$\frac{dy}{dx} = -7x^{-7-1}$ $\frac{dy}{dx} = -7x^{-8}$
The final answer for 4(4) is $\boxed{-7x^{-8}}$ .

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