Find dy/dx y=1/3(x^2+2)^(3/2)

Solution:

Step 1

Apply the derivative operator to both sides of the equation: $\frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{1}{3}(x^2 + 2)^{\frac{3}{2}}\right)$.

Step 2

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step 3

Compute the derivative of the right-hand side.

Step 3.1

The constant $\frac{1}{3}$ remains unchanged when differentiating: $\frac{1}{3}\frac{d}{dx}[(x^2 + 2)^{\frac{3}{2}}]$.

Step 3.2

Apply the chain rule for derivatives: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = x^{\frac{3}{2}}$ and $g(x) = x^2 + 2$.

Step 3.2.1

Introduce a substitution $u = x^2 + 2$: $\frac{1}{3}\left(\frac{d}{du}[u^{\frac{3}{2}}]\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.2.2

Differentiate $u^{\frac{3}{2}}$ using the power rule: $\frac{1}{3}\left(\frac{3}{2}u^{\frac{3}{2} - 1}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.2.3

Substitute $u$ back with $x^2 + 2$: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{3}{2} - 1}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.3

Express $-1$ as a fraction: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{3}{2} - 1 \cdot \frac{2}{2}}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.4

Combine the exponents: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{3}{2} + \frac{-1 \cdot 2}{2}}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.5

Simplify the exponent: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{3 - 1 \cdot 2}{2}}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.6

Simplify the numerator.

Step 3.6.1

Multiply $-1$ by $2$: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{3 - 2}{2}}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.6.2

Subtract $2$ from $3$: $\frac{1}{3}\left(\frac{3}{2}(x^2 + 2)^{\frac{1}{2}}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.7

Combine the constants: $\frac{1}{3}\left(\frac{3(x^2 + 2)^{\frac{1}{2}}}{2}\frac{d}{dx}[x^2 + 2]\right)$.

Step 3.8

Multiply the constants: $\frac{3(x^2 + 2)^{\frac{1}{2}}}{2 \cdot 3}\frac{d}{dx}[x^2 + 2]$.

Step 3.9

Multiply the denominators: $\frac{3(x^2 + 2)^{\frac{1}{2}}}{6}\frac{d}{dx}[x^2 + 2]$.

Step 3.10

Factor out the constant: $\frac{3((x^2 + 2)^{\frac{1}{2}})}{6}\frac{d}{dx}[x^2 + 2]$.

Step 3.11

Cancel the common factors.

Step 3.11.1

Factor out the constant: $\frac{3(x^2 + 2)^{\frac{1}{2}}}{3 \cdot 2}\frac{d}{dx}[x^2 + 2]$.

Step 3.11.2

Cancel the common factor: $\frac{\cancel{3}(x^2 + 2)^{\frac{1}{2}}}{\cancel{3} \cdot 2}\frac{d}{dx}[x^2 + 2]$.

Step 3.11.3

Rewrite the expression: $\frac{(x^2 + 2)^{\frac{1}{2}}}{2}\frac{d}{dx}[x^2 + 2]$.

Step 3.12

Apply the sum rule: $\frac{d}{dx}[x^2 + 2] = \frac{d}{dx}[x^2] + \frac{d}{dx}[2]$.

Step 3.13

Differentiate $x^2$ using the power rule: $\frac{(x^2 + 2)^{\frac{1}{2}}}{2}(2x + \frac{d}{dx}[2])$.

Step 3.14

The derivative of a constant is zero: $\frac{(x^2 + 2)^{\frac{1}{2}}}{2}(2x + 0)$.

Step 3.15

Simplify the expression.

Step 3.15.1

Combine terms: $\frac{(x^2 + 2)^{\frac{1}{2}}}{2}(2x)$.

Step 3.15.2

Combine the constants: $\frac{2(x^2 + 2)^{\frac{1}{2}}}{2}x$.

Step 3.15.3

Multiply the terms: $\frac{2(x^2 + 2)^{\frac{1}{2}}x}{2}$.

Step 3.15.4

Cancel the common factor: $\frac{\cancel{2}(x^2 + 2)^{\frac{1}{2}}x}{\cancel{2}}$.

Step 3.15.5

Finalize the simplification.

Step 3.15.5.1

Divide by $1$: $(x^2 + 2)^{\frac{1}{2}}x$.

Step 3.15.5.2

Reorder the factors: $x(x^2 + 2)^{\frac{1}{2}}$.

Step 4

Combine the left and right sides: $y = x(x^2 + 2)^{\frac{1}{2}}$.

Step 5

Replace $y$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = x(x^2 + 2)^{\frac{1}{2}}$.

Knowledge Notes:

1. Derivative Operator: The notation $\frac{d}{dx}$ represents the derivative of a function with respect to the variable $x$.

2. Chain Rule: A fundamental rule in calculus for differentiating compositions of functions. It states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$.

3. Power Rule: A basic rule for differentiation. If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

4. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.

5. Constant Rule: The derivative of a constant is zero.

6. Simplification: In calculus, simplifying expressions involves canceling common factors, combining like terms, and applying arithmetic operations to make the expression as concise as possible.