Question 2: a) Y = 4x³ + 5x² − 9x + 4, differentiate y with respect to x. b) y = (3x + 5)/(6x − 9)…

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Question 2: a) Y = 4x³ + 5x² − 9x + 4, differentiate y with respect to x. b) y = (3x + 5)/(6x − 9). Differentiate y with respect to x.

ScanSolve AI Tutor · Accepted Answer

a) To differentiate Y = 4x³ + 5x² − 9x + 4 with respect to x: Step 1: Apply the power rule for differentiation, d/dx(axⁿ) = anxⁿ⁻¹, to each term. The derivative of a constant is 0. dY/dx = d/dx(4x³) + d/dx(5x²) − d/dx(9x) + d/dx(4) Step 2: Differentiate each term. dY/dx = (4 · 3x³⁻¹) + (5 · 2x²⁻¹) − (9 · 1x¹⁻¹) + 0; dY/dx = 12x² + 10x¹ − 9x⁰ Step 3: Simplify the expression. dY/dx = 12x² + 10x − 9 The derivative is 12x² + 10x − 9. b) To differentiate y = (3x+5)/(6x−9) with respect to x: Step 1: Use the quotient rule, which states that if y = u/v, then dy/dx = (v du/dx − u dv/dx)/(v²). Let u = 3x+5 and v = 6x−9. Step 2: Find the derivatives of u and v. du/dx = d/dx(3x+5) = 3; dv/dx = d/dx(6x−9) = 6 Step 3: Substitute u, v, du/dx, dv/dx into the quotient rule formula. dy/dx = ((6x−9)(3) − (3x+5)(6))/((6x−9)²) Step 4: Expand and simplify the numerator. dy/dx = ((18x − 27) − (18x + 30))/((6x−9)²); dy/dx = (18x − 27 − 18x − 30)/((6x−9)²); dy/dx = (−57)/((6x−9)²) The derivative is (−57)/((6x−9)²).

Question 2: a) Y = 4x³ + 5x² − 9x + 4, differentiate y with respect to x. b) y = (3x + 5)/(6x - 9). Differentiate y with respect to x.

Answers

Step-by-step solution

Part a· 3 steps

Apply the power rule for differentiation, $\frac{d}{dx}(ax^n) = anx^{n-1}$ , to each term.

Differentiate each term.

Simplify the expression.

Part b· 4 steps
$\frac{-57}{(6x-9)^2}$

Use the quotient rule, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, \frac{du}{dx}, \frac{dv}{dx}$ into the quotient rule formula.

Expand and simplify the numerator.

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