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

Question

ScanSolve AI Tutor · Accepted Answer

Here is the solution for question 4 from the image.

4. Differentiate the following

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

Apply the power rule for differentiation, $\frac{d}{dx}(ax^n) = anx^{n-1}$ , and the sum/difference rule.
$\frac{dy}{dx} = \frac{d}{dx}(7x^4) + \frac{d}{dx}(5x^3) - \frac{d}{dx}(7x)$
Differentiate each term.
$\frac{d}{dx}(7x^4) = 7 \cdot 4x^{4-1} = 28x^3$ $\frac{d}{dx}(5x^3) = 5 \cdot 3x^{3-1} = 15x^2$ $\frac{d}{dx}(7x) = 7 \cdot 1x^{1-1} = 7x^0 = 7$
Combine the derivatives.
$\frac{dy}{dx} = 28x^3 + 15x^2 - 7$
The final answer for 4(1) is $\boxed{28x^3 + 15x^2 - 7}$ .

(2) $y = x^3$
Apply the power rule for differentiation, $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$\frac{dy}{dx} = 3x^{3-1}$
Simplify the exponent.
$\frac{dy}{dx} = 3x^2$
The final answer for 4(2) is $\boxed{3x^2}$ .

(3) $y = 8x^2 - x + 5x$
Simplify the expression for $y$ .
$y = 8x^2 + (-x + 5x)$ $y = 8x^2 + 4x$
Apply the power rule for differentiation, $\frac{d}{dx}(ax^n) = anx^{n-1}$ , and the sum rule.
$\frac{dy}{dx} = \frac{d}{dx}(8x^2) + \frac{d}{dx}(4x)$
Differentiate each term.
$\frac{d}{dx}(8x^2) = 8 \cdot 2x^{2-1} = 16x^1 = 16x$ $\frac{d}{dx}(4x) = 4 \cdot 1x^{1-1} = 4x^0 = 4$
Combine the derivatives.
$\frac{dy}{dx} = 16x + 4$
The final answer for 4(3) is $\boxed{16x + 4}$ .

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

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