Math / Calculus

QuestionDifferentiate the following function. $y=\frac{3 \ln (4 x)}{\sqrt{x}}$
$\frac{d}{d x}\left(\frac{3 \ln (4 x)}{\sqrt{x}}\right)=$ $\square$

Studdy Solution

STEP 1

1. We are given the function $y = \frac{3 \ln (4x)}{\sqrt{x}}$ .
2. We need to differentiate this function with respect to $x$ .

STEP 2

1. Rewrite the function in a form that is easier to differentiate.
2. Apply the quotient rule for differentiation.
3. Simplify the derivative expression.

STEP 3

Rewrite the function using exponent rules:
$y = 3 \ln (4x) \cdot x^{-1/2}$

STEP 4

Apply the quotient rule. The quotient rule states that if $y = \frac{u}{v}$ , then:
$\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
In this case, let $u = 3 \ln (4x)$ and $v = \sqrt{x} = x^{1/2}$ .

STEP 5

Differentiate $u = 3 \ln (4x)$ with respect to $x$ :
$\frac{du}{dx} = 3 \cdot \frac{1}{4x} \cdot 4 = \frac{3}{x}$
Differentiate $v = x^{1/2}$ with respect to $x$ :
$\frac{dv}{dx} = \frac{1}{2} x^{-1/2}$

STEP 6

Substitute into the quotient rule formula:
$\frac{dy}{dx} = \frac{x^{1/2} \cdot \frac{3}{x} - 3 \ln (4x) \cdot \frac{1}{2} x^{-1/2}}{(x^{1/2})^2}$
Simplify the expression:
$\frac{dy}{dx} = \frac{x^{-1/2} \cdot 3 - 3 \ln (4x) \cdot \frac{1}{2} x^{-1/2}}{x}$
$\frac{dy}{dx} = \frac{3x^{-1/2} - \frac{3}{2} \ln (4x) x^{-1/2}}{x}$
$\frac{dy}{dx} = x^{-3/2} \left(3 - \frac{3}{2} \ln (4x)\right)$
The derivative of the function is:
$\boxed{x^{-3/2} \left(3 - \frac{3}{2} \ln (4x)\right)}$

Was this helpful?

QuestionDifferentiate the following function. $y=\frac{3 \ln (4 x)}{\sqrt{x}}$
$\frac{d}{d x}\left(\frac{3 \ln (4 x)}{\sqrt{x}}\right)=$ $\square$

Studdy Solution

STEP 1

1. We are given the function $y = \frac{3 \ln (4x)}{\sqrt{x}}$ .
2. We need to differentiate this function with respect to $x$ .

STEP 2

1. Rewrite the function in a form that is easier to differentiate.
2. Apply the quotient rule for differentiation.
3. Simplify the derivative expression.

STEP 3

Rewrite the function using exponent rules:
$y = 3 \ln (4x) \cdot x^{-1/2}$

STEP 4

Apply the quotient rule. The quotient rule states that if $y = \frac{u}{v}$ , then:
$\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
In this case, let $u = 3 \ln (4x)$ and $v = \sqrt{x} = x^{1/2}$ .

STEP 5

Differentiate $u = 3 \ln (4x)$ with respect to $x$ :
$\frac{du}{dx} = 3 \cdot \frac{1}{4x} \cdot 4 = \frac{3}{x}$
Differentiate $v = x^{1/2}$ with respect to $x$ :
$\frac{dv}{dx} = \frac{1}{2} x^{-1/2}$

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionDifferentiate the following function. y=3ln⁡(4x)xy=\frac{3 \ln (4 x)}{\sqrt{x}}y=x​3ln(4x)​ddx(3ln⁡(4x)x)=\frac{d}{d x}\left(\frac{3 \ln (4 x)}{\sqrt{x}}\right)=dxd​(x​3ln(4x)​)= □\square□

Studdy Solution

STEP 1

1. We are given the function y=3ln⁡(4x)x y = \frac{3 \ln (4x)}{\sqrt{x}} y=x​3ln(4x)​.2. We need to differentiate this function with respect to x x x.

STEP 2

1. Rewrite the function in a form that is easier to differentiate.2. Apply the quotient rule for differentiation.3. Simplify the derivative expression.

STEP 3

Rewrite the function using exponent rules:y=3ln⁡(4x)⋅x−1/2 y = 3 \ln (4x) \cdot x^{-1/2} y=3ln(4x)⋅x−1/2

STEP 4

STEP 5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionDifferentiate the following function. $y=\frac{3 \ln (4 x)}{\sqrt{x}}$
$\frac{d}{d x}\left(\frac{3 \ln (4 x)}{\sqrt{x}}\right)=$ $\square$

1. We are given the function $y = \frac{3 \ln (4x)}{\sqrt{x}}$ .
2. We need to differentiate this function with respect to $x$ .

1. Rewrite the function in a form that is easier to differentiate.
2. Apply the quotient rule for differentiation.
3. Simplify the derivative expression.

Rewrite the function using exponent rules:
$y = 3 \ln (4x) \cdot x^{-1/2}$