Find the derivatives of the following functions: y = 4x4 + 3x3 + 2x2 - 2x − 1 and F(x) = x3 - x2 - x + 1.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the derivatives of the given functions.

For the first function:

Write the given function.
$y = 4x^4 + 3x^3 + 2x^2 - 2x - 1$
Differentiate each term using the power rule, $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$\frac{dy}{dx} = \frac{d}{dx}(4x^4) + \frac{d}{dx}(3x^3) + \frac{d}{dx}(2x^2) - \frac{d}{dx}(2x) - \frac{d}{dx}(1)$
Apply the power rule to each term.
$\frac{dy}{dx} = (4 \cdot 4x^{4-1}) + (3 \cdot 3x^{3-1}) + (2 \cdot 2x^{2-1}) - (2 \cdot 1x^{1-1}) - 0$
Simplify the expression.
$\frac{dy}{dx} = 16x^3 + 9x^2 + 4x - 2$
The derivative of the first function is:
$\boxed{\frac{dy}{dx} = 16x^3 + 9x^2 + 4x - 2}$
For the second function:
Write the given function.
$F(x) = x^3 - x^2 - x + 1$
Differentiate each term using the power rule, $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$F'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(x^2) - \frac{d}{dx}(x) + \frac{d}{dx}(1)$
Apply the power rule to each term.
$F'(x) = (1 \cdot 3x^{3-1}) - (1 \cdot 2x^{2-1}) - (1 \cdot 1x^{1-1}) + 0$
Simplify the expression.
$F'(x) = 3x^2 - 2x - 1$
The derivative of the second function is:
$\boxed{F'(x) = 3x^2 - 2x - 1}$

Find the derivatives of the following functions: y = 4x4 + 3x3 + 2x2 − 2x − 1 and F(x) = x3 − x2 − x + 1.