Answer :
To find [tex]\(\frac{dy}{dx}\)[/tex] using the given functions [tex]\(y = f(u)\)[/tex] and [tex]\(u = g(x)\)[/tex], we'll apply the chain rule in Leibniz's notation. The chain rule states that:
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
We are given the functions:
[tex]\[ y = 5u^4 + 2 \][/tex]
[tex]\[ u = 3x^3 \][/tex]
Let's break down the solution step-by-step:
### Step 1: Find [tex]\(\frac{dy}{du}\)[/tex]
First, we need to find the derivative of [tex]\(y\)[/tex] with respect to [tex]\(u\)[/tex]:
[tex]\[ \frac{dy}{du} = \frac{d}{du}(5u^4 + 2) \][/tex]
Differentiate term by term:
[tex]\[ \frac{dy}{du} = 20u^3 \][/tex]
### Step 2: Find [tex]\(\frac{du}{dx}\)[/tex]
Next, we need to find the derivative of [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx}(3x^3) \][/tex]
Differentiate:
[tex]\[ \frac{du}{dx} = 9x^2 \][/tex]
### Step 3: Apply the Chain Rule
Now that we have both derivatives, we can apply the chain rule:
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
Substitute [tex]\(\frac{dy}{du}\)[/tex] and [tex]\(\frac{du}{dx}\)[/tex]:
[tex]\[ \frac{dy}{dx} = (20u^3) \cdot (9x^2) \][/tex]
### Step 4: Substitute [tex]\(u\)[/tex]
Recall that [tex]\(u = 3x^3\)[/tex]. Let's substitute [tex]\(u\)[/tex] with [tex]\(3x^3\)[/tex]:
[tex]\[ \frac{dy}{dx} = 20(3x^3)^3 \cdot 9x^2 \][/tex]
### Step 5: Compute with Given [tex]\(x = 1\)[/tex]
Let's compute the values at [tex]\(x = 1\)[/tex]:
[tex]\[ u = 3(1)^3 = 3 \][/tex]
Calculate [tex]\(\frac{dy}{du}\)[/tex] at [tex]\(u = 3\)[/tex]:
[tex]\[ \frac{dy}{du} = 20(3)^3 = 20 \cdot 27 = 540 \][/tex]
Calculate [tex]\(\frac{du}{dx}\)[/tex] at [tex]\(x = 1\)[/tex]:
[tex]\[ \frac{du}{dx} = 9(1)^2 = 9 \][/tex]
### Step 6: Final Calculation
Now, multiply the derivatives:
[tex]\[ \frac{dy}{dx} = 540 \cdot 9 = 4860 \][/tex]
Thus, the result of [tex]\(\frac{dy}{dx}\)[/tex] is 4860.
Summary of Numerical Results:
[tex]\[ u = 3, \quad \frac{dy}{du} = 540, \quad \frac{du}{dx} = 9, \quad \frac{dy}{dx} = 4860. \][/tex]
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
We are given the functions:
[tex]\[ y = 5u^4 + 2 \][/tex]
[tex]\[ u = 3x^3 \][/tex]
Let's break down the solution step-by-step:
### Step 1: Find [tex]\(\frac{dy}{du}\)[/tex]
First, we need to find the derivative of [tex]\(y\)[/tex] with respect to [tex]\(u\)[/tex]:
[tex]\[ \frac{dy}{du} = \frac{d}{du}(5u^4 + 2) \][/tex]
Differentiate term by term:
[tex]\[ \frac{dy}{du} = 20u^3 \][/tex]
### Step 2: Find [tex]\(\frac{du}{dx}\)[/tex]
Next, we need to find the derivative of [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx}(3x^3) \][/tex]
Differentiate:
[tex]\[ \frac{du}{dx} = 9x^2 \][/tex]
### Step 3: Apply the Chain Rule
Now that we have both derivatives, we can apply the chain rule:
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
Substitute [tex]\(\frac{dy}{du}\)[/tex] and [tex]\(\frac{du}{dx}\)[/tex]:
[tex]\[ \frac{dy}{dx} = (20u^3) \cdot (9x^2) \][/tex]
### Step 4: Substitute [tex]\(u\)[/tex]
Recall that [tex]\(u = 3x^3\)[/tex]. Let's substitute [tex]\(u\)[/tex] with [tex]\(3x^3\)[/tex]:
[tex]\[ \frac{dy}{dx} = 20(3x^3)^3 \cdot 9x^2 \][/tex]
### Step 5: Compute with Given [tex]\(x = 1\)[/tex]
Let's compute the values at [tex]\(x = 1\)[/tex]:
[tex]\[ u = 3(1)^3 = 3 \][/tex]
Calculate [tex]\(\frac{dy}{du}\)[/tex] at [tex]\(u = 3\)[/tex]:
[tex]\[ \frac{dy}{du} = 20(3)^3 = 20 \cdot 27 = 540 \][/tex]
Calculate [tex]\(\frac{du}{dx}\)[/tex] at [tex]\(x = 1\)[/tex]:
[tex]\[ \frac{du}{dx} = 9(1)^2 = 9 \][/tex]
### Step 6: Final Calculation
Now, multiply the derivatives:
[tex]\[ \frac{dy}{dx} = 540 \cdot 9 = 4860 \][/tex]
Thus, the result of [tex]\(\frac{dy}{dx}\)[/tex] is 4860.
Summary of Numerical Results:
[tex]\[ u = 3, \quad \frac{dy}{du} = 540, \quad \frac{du}{dx} = 9, \quad \frac{dy}{dx} = 4860. \][/tex]
