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: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$, 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: $\frac{d}{dx}{x}^n=n{x}^{n-1}$, 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 $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

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 $f^{\prime }(x)=n{x}^{n-1}$.

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.