Given the function y = 2x5 + 4x4 - x3 + 3x2 - 5x + 7, find the value of dy/dx when x=2.

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Step 1: The given function is: y = 2x⁵ + 4x⁴ − x³ + 3x² − 5x + 7; Step 2: Differentiate each term of the polynomial with respect to x using the power rule d/dx(axⁿ) = anxⁿ⁻¹ and the rule that the derivative of a constant is 0. dy/dx = d/dx(2x⁵) + d/dx(4x⁴) − d/dx(x³) + d/dx(3x²) − d/dx(5x) + d/dx(7); dy/dx = (2 · 5)x⁵⁻¹ + (4 · 4)x⁴⁻¹ − (1 · 3)x³⁻¹ + (3 · 2)x²⁻¹ − (5 · 1)x¹⁻¹ + 0 Step 3: Simplify the expression for dy/dx. dy/dx = 10x⁴ + 16x³ − 3x² + 6x − 5; Step 4: Substitute x=2 into the expression for dy/dx to find its value. dy/dx |_x=2 = 10(2)⁴ + 16(2)³ − 3(2)² + 6(2) − 5; dy/dx |_x=2 = 10(16) + 16(8) − 3(4) + 12 − 5; dy/dx |_x=2 = 160 + 128 − 12 + 12 − 5 Step 5: Perform the arithmetic operations. dy/dx |_x=2 = 288 − 12 + 12 − 5; dy/dx |_x=2 = 276 + 12 − 5; dy/dx |_x=2 = 288 − 5; dy/dx |_x=2 = 283

Point	x	y
x-intercept	−2.62	0
Maximum	−1.97	37.04
y-intercept	0	7
Minimum	0.51	5.44

Given the function y = 2x5 + 4x4 - x3 + 3x2 - 5x + 7, find the value of dy/dx when x=2.

Answer

Step-by-step solution

The given function is

Differentiate each term of the polynomial with respect to $x$ using the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ and the rule that the derivative of a constant is 0.

Simplify the expression for $\frac{dy}{dx}$ .

Substitute $x=2$ into the expression for $\frac{dy}{dx}$ to find its value.

Perform the arithmetic operations.

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Given the function y = 2x5 + 4x4 - x3 + 3x2 - 5x + 7, find the value of dy/dx when x=2.

Answer

Step-by-step solution

The given function is

Differentiate each term of the polynomial with respect to xxx using the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1 and the rule that the derivative of a constant is 0.

Simplify the expression for dydx\frac{dy}{dx}dxdy​.

Substitute x=2x=2x=2 into the expression for dydx\frac{dy}{dx}dxdy​ to find its value.

Perform the arithmetic operations.

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More Mathematics Questions

Differentiate each term of the polynomial with respect to $x$ using the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ and the rule that the derivative of a constant is 0.

Simplify the expression for $\frac{dy}{dx}$ .

Substitute $x=2$ into the expression for $\frac{dy}{dx}$ to find its value.