Math / Calculus

QuestionFind the derivative of the function $y=\sqrt{5+3 x}$ . $\frac{d y}{d x}=$

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of a square root function with a little something extra inside. Watch out! Don't forget about the chain rule, and remember how to differentiate a square root!

STEP 2

1. Rewrite the Function
2. Apply the Chain Rule
3. Simplify the Derivative

STEP 3

Let's rewrite our function $y = \sqrt{5 + 3x}$ using a fractional exponent.
This makes it easier to apply the power rule for differentiation.
Remember, a square root is the same as raising to the power of $\frac{1}{2}$ .
So, we can rewrite the function as $y = (5 + 3x)^{\frac{1}{2}}$ .
This sets us up perfectly for the next step!

STEP 4

Now, it's chain rule time!
Remember, the chain rule says the derivative of a composite function is the derivative of the outer function (with the inside function left alone) times the derivative of the inside function.
Our outer function is raising something to the power of $\frac{1}{2}$ , and our inner function is $5 + 3x$ .

STEP 5

Let's differentiate the outer function first.
Using the power rule, we bring the exponent $\frac{1}{2}$ down in front and reduce the exponent by 1: $\frac{1}{2}(5 + 3x)^{\frac{1}{2} - 1}$ .
This simplifies to $\frac{1}{2}(5 + 3x)^{-\frac{1}{2}}$ .

STEP 6

Next, we differentiate the inner function, $5 + 3x$ .
The derivative of 5 (a constant) is 0, and the derivative of $3x$ is just 3.
So, the derivative of the inner function is simply 3.

STEP 7

Finally, we multiply the results from differentiating the outer and inner functions together: $\frac{1}{2}(5 + 3x)^{-\frac{1}{2}} \cdot 3$ .

STEP 8

Let's make our derivative look a little nicer.
We can rewrite $(5 + 3x)^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{5 + 3x}}$ .
So, our derivative becomes $\frac{1}{2} \cdot \frac{1}{\sqrt{5 + 3x}} \cdot 3$ .

STEP 9

Multiplying the fractions, we get $\frac{3}{2\sqrt{5 + 3x}}$ .
This is our final, simplified derivative!

STEP 10

$\frac{dy}{dx} = \frac{3}{2\sqrt{5 + 3x}}$

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QuestionFind the derivative of the function $y=\sqrt{5+3 x}$ . $\frac{d y}{d x}=$

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of a square root function with a little something extra inside. Watch out! Don't forget about the chain rule, and remember how to differentiate a square root!

STEP 2

1. Rewrite the Function
2. Apply the Chain Rule
3. Simplify the Derivative

STEP 3

STEP 4

STEP 5

Let's differentiate the outer function first.
Using the power rule, we bring the exponent $\frac{1}{2}$ down in front and reduce the exponent by 1: $\frac{1}{2}(5 + 3x)^{\frac{1}{2} - 1}$ .
This simplifies to $\frac{1}{2}(5 + 3x)^{-\frac{1}{2}}$ .

STEP 6

Next, we differentiate the inner function, $5 + 3x$ .
The derivative of 5 (a constant) is 0, and the derivative of $3x$ is just 3.
So, the derivative of the inner function is simply 3.

STEP 7

Finally, we multiply the results from differentiating the outer and inner functions together: $\frac{1}{2}(5 + 3x)^{-\frac{1}{2}} \cdot 3$ .

STEP 8

Let's make our derivative look a little nicer.
We can rewrite $(5 + 3x)^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{5 + 3x}}$ .
So, our derivative becomes $\frac{1}{2} \cdot \frac{1}{\sqrt{5 + 3x}} \cdot 3$ .

STEP 9

Multiplying the fractions, we get $\frac{3}{2\sqrt{5 + 3x}}$ .
This is our final, simplified derivative!

STEP 10

$\frac{dy}{dx} = \frac{3}{2\sqrt{5 + 3x}}$

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QuestionFind the derivative of the function y=5+3xy=\sqrt{5+3 x}y=5+3x​. dydx=\frac{d y}{d x}=dxdy​=

Studdy Solution

STEP 1

What is this asking? We need to find the derivative of a square root function with a little something extra inside. Watch out! Don't forget about the chain rule, and remember how to differentiate a square root!

STEP 2

1. Rewrite the Function2. Apply the Chain Rule3. Simplify the Derivative

STEP 3

STEP 4

STEP 5

STEP 6

Next, we **differentiate** the inner function, 5+3x5 + 3x5+3x.The derivative of 5 (a constant) is 0, and the derivative of 3x3x3x is just 3.So, the derivative of the inner function is simply 3.

STEP 7

Finally, we **multiply** the results from differentiating the outer and inner functions together: 12(5+3x)−12⋅3\frac{1}{2}(5 + 3x)^{-\frac{1}{2}} \cdot 321​(5+3x)−21​⋅3.

STEP 8

Let's make our **derivative** look a little nicer.We can rewrite (5+3x)−12(5 + 3x)^{-\frac{1}{2}}(5+3x)−21​ as 15+3x\frac{1}{\sqrt{5 + 3x}}5+3x​1​.So, our derivative becomes 12⋅15+3x⋅3\frac{1}{2} \cdot \frac{1}{\sqrt{5 + 3x}} \cdot 321​⋅5+3x​1​⋅3.

STEP 9

Multiplying the fractions, we get 325+3x\frac{3}{2\sqrt{5 + 3x}}25+3x​3​.This is our **final, simplified derivative**!

STEP 10

dydx=325+3x\frac{dy}{dx} = \frac{3}{2\sqrt{5 + 3x}}dxdy​=25+3x​3​

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QuestionFind the derivative of the function $y=\sqrt{5+3 x}$ . $\frac{d y}{d x}=$

1. Rewrite the Function
2. Apply the Chain Rule
3. Simplify the Derivative

Next, we differentiate the inner function, $5 + 3x$ .
The derivative of 5 (a constant) is 0, and the derivative of $3x$ is just 3.
So, the derivative of the inner function is simply 3.

Finally, we multiply the results from differentiating the outer and inner functions together: $\frac{1}{2}(5 + 3x)^{-\frac{1}{2}} \cdot 3$ .

Let's make our derivative look a little nicer.
We can rewrite $(5 + 3x)^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{5 + 3x}}$ .
So, our derivative becomes $\frac{1}{2} \cdot \frac{1}{\sqrt{5 + 3x}} \cdot 3$ .

Multiplying the fractions, we get $\frac{3}{2\sqrt{5 + 3x}}$ .
This is our final, simplified derivative!

$\frac{dy}{dx} = \frac{3}{2\sqrt{5 + 3x}}$