Solved on Feb 06, 2024

Find the derivative of y=4x2+2xy=4x^2+2\sqrt{x} evaluated at x=1x=1.

STEP 1

Assumptions
1. The given function is y=4x2+2x y = 4x^2 + 2\sqrt{x} .
2. We need to find the derivative of y y with respect to x x , denoted as dydx \frac{dy}{dx} .
3. After finding the derivative, we will evaluate dydx \frac{dy}{dx} at x=1 x = 1 .

STEP 2

To find the derivative of y y with respect to x x , we will use the power rule for differentiation. The power rule states that if y=axn y = ax^n , then dydx=naxn1 \frac{dy}{dx} = n \cdot ax^{n-1} .

STEP 3

The term 4x2 4x^2 can be differentiated using the power rule. Applying the power rule to this term gives us:
ddx(4x2)=42x21=8x \frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x

STEP 4

The term 2x 2\sqrt{x} can be rewritten as 2x1/2 2x^{1/2} to make it easier to differentiate using the power rule.

STEP 5

Now, apply the power rule to 2x1/2 2x^{1/2} :
ddx(2x1/2)=212x1/21=x1/2 \frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2}

STEP 6

Simplify the derivative of the second term:
x1/2=1x x^{-1/2} = \frac{1}{\sqrt{x}}

STEP 7

Combine the derivatives of both terms to get the derivative of y y :
dydx=8x+1x \frac{dy}{dx} = 8x + \frac{1}{\sqrt{x}}

STEP 8

Now, we need to evaluate dydx \frac{dy}{dx} at x=1 x = 1 . Substitute x=1 x = 1 into the derivative.
dydxx=1=8(1)+11 \frac{dy}{dx}\bigg|_{x=1} = 8(1) + \frac{1}{\sqrt{1}}

STEP 9

Simplify the expression by evaluating the square root of 1 and the multiplication:
dydxx=1=8+1 \frac{dy}{dx}\bigg|_{x=1} = 8 + 1

STEP 10

Add the two numbers to find the value of the derivative at x=1 x = 1 :
dydxx=1=9 \frac{dy}{dx}\bigg|_{x=1} = 9
The derivative of y y with respect to x x when x=1 x = 1 is 9.
The correct answer is: 9

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