Find dy/dx y=3^( square root of x)

Solution:

Step 1:

Rewrite the given function using the property that $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. Thus, we have $y = 3^{x^{\frac{1}{2}}}$.

Step 2:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}y = \frac{d}{dx}3^{x^{\frac{1}{2}}}$.

Step 3:

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

Step 4:

Proceed to differentiate the right-hand side of the equation.

Step 4.1:

Apply the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = 3^{x}$ and $g(x) = x^{\frac{1}{2}}$.

Step 4.1.1:

Let $u = x^{\frac{1}{2}}$. Then differentiate $3^{u}$ with respect to $u$ and $x^{\frac{1}{2}}$ with respect to $x$: $\frac{d}{du}3^{u} \cdot \frac{d}{dx}x^{\frac{1}{2}}$.

Step 4.1.2:

Use the exponential rule for differentiation: $\frac{d}{du}a^{u} = a^{u} \ln(a)$, where $a = 3$. This gives us $3^{u} \ln(3) \cdot \frac{d}{dx}x^{\frac{1}{2}}$.

Step 4.1.3:

Substitute back $u = x^{\frac{1}{2}}$ to get $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{d}{dx}x^{\frac{1}{2}}$.

Step 4.2:

Apply the power rule for differentiation: $\frac{d}{dx}x^{n} = nx^{n-1}$, where $n = \frac{1}{2}$. This results in $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{\frac{1}{2} - 1}$.

Step 4.3:

Express $-1$ as a fraction with a common denominator by multiplying by $\frac{2}{2}$: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$.

Step 4.4:

Combine the terms in the exponent: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{\frac{1}{2} + \frac{-2}{2}}$.

Step 4.5:

Add the numerators over the common denominator: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{\frac{1 - 2}{2}}$.

Step 4.6:

Simplify the exponent by performing the subtraction in the numerator: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{\frac{-1}{2}}$.

Step 4.7:

Rewrite the expression by moving the negative exponent to the denominator: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{1}{2} x^{-\frac{1}{2}}$.

Step 4.8:

Combine the fraction and the exponent: $3^{x^{\frac{1}{2}}} \ln(3) \cdot \frac{x^{-\frac{1}{2}}}{2}$.

Step 4.9:

Multiply the terms together: $\frac{x^{-\frac{1}{2}} \cdot 3^{x^{\frac{1}{2}}}}{2} \ln(3)$.

Step 4.10:

Combine the fraction and the natural logarithm: $\frac{x^{-\frac{1}{2}} \cdot 3^{x^{\frac{1}{2}}} \ln(3)}{2}$.

Step 4.11:

Apply the negative exponent rule $b^{-n} = \frac{1}{b^{n}}$ to move $x^{-\frac{1}{2}}$ to the denominator: $\frac{3^{x^{\frac{1}{2}}} \ln(3)}{2x^{\frac{1}{2}}}$.

Step 5:

Express the derivative $\frac{dy}{dx}$ as the result of the differentiation: $\frac{dy}{dx} = \frac{3^{x^{\frac{1}{2}}} \ln(3)}{2x^{\frac{1}{2}}}$.

Step 6:

Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx} = \frac{3^{x^{\frac{1}{2}}} \ln(3)}{2x^{\frac{1}{2}}}$.

Knowledge Notes:

1. Exponentiation of a Root: The expression $\sqrt[n]{a^{x}}$ can be rewritten as $a^{\frac{x}{n}}$.

2. Chain Rule: The chain rule is a fundamental property in calculus used to differentiate compositions of functions. It states that $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

3. Exponential Rule: When differentiating an exponential function $a^{u}$, where $a$ is a constant and $u$ is a function of $x$, the derivative is $a^{u} \ln(a) \cdot \frac{du}{dx}$.

4. Power Rule: The power rule for differentiation states that if $f(x) = x^{n}$, then $f'(x) = nx^{n-1}$.

5. Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^{n}}$.

6. Natural Logarithm (ln): The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828.