Find dy/dx y = seventh root of x

Solution:

Step 1:

Transform the expression $\sqrt[7]{x}$ into exponential form as $x^{\frac{1}{7}}$. Thus, we have $y=x^{\frac{1}{7}}$.

Step 2:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y)=\frac{d}{dx}\left(x^{\frac{1}{7}}\right)$.

Step 3:

The derivative of $y$ in terms of $x$ is denoted as $\frac{dy}{dx}$.

Step 4:

Apply the differentiation to the right-hand side.

Step 4.1:

Employ the Power Rule for differentiation, which is $\frac{d}{dx}(x^n)=n{x}^{n-1}$, where $n=\frac{1}{7}$, to obtain $\frac{1}{7}{x}^{\frac{1}{7}-1}$.

Step 4.2:

Express $-1$ as a fraction with the same denominator by multiplying it by $\frac{7}{7}$, resulting in $\frac{1}{7}{x}^{\frac{1}{7}-1\cdot \frac{7}{7}}$.

Step 4.3:

Combine the terms $-1$ and $\frac{7}{7}$ to get $\frac{1}{7}{x}^{\frac{1}{7}-\frac{7}{7}}$.

Step 4.4:

Add the numerators over the common denominator to get $\frac{1}{7}{x}^{\frac{1-7}{7}}$.

Step 4.5:

Simplify the numerator.

Step 4.5.1:

Calculate $-1$ times $7$ to get $\frac{1}{7}{x}^{\frac{1-7}{7}}$.

Step 4.5.2:

Subtract $7$ from $1$, which simplifies to $\frac{1}{7}{x}^{\frac{-6}{7}}$.

Step 4.6:

Position the negative exponent in front of the fraction to obtain $\frac{1}{7}{x}^{-\frac{6}{7}}$.

Step 4.7:

Simplify the expression further.

Step 4.7.1:

Apply the negative exponent rule, which states $b^{-n}=\frac{1}{b^n}$, to rewrite the expression as $\frac{1}{7}\cdot \frac{1}{x^{\frac{6}{7}}}$.

Step 4.7.2:

Multiply $\frac{1}{7}$ by $\frac{1}{x^{\frac{6}{7}}}$ to get $\frac{1}{7x^{\frac{6}{7}}}$.

Step 5:

Combine the left and right sides to form the equation $y=\frac{1}{7x^{\frac{6}{7}}}$.

Step 6:

Substitute $\frac{dy}{dx}$ for $y$ to obtain the final derivative $\frac{dy}{dx}=\frac{1}{7x^{\frac{6}{7}}}$.

Knowledge Notes:

1. Exponential Form of a Root: The nth root of a number a, denoted as $\sqrt[n]{a}$, can be expressed in exponential form as $a^{\frac{1}{n}}$.

2. Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{dx}$ represents the derivative with respect to $x$.

3. Power Rule for Differentiation: A fundamental rule for finding derivatives, the Power Rule states that if $f(x)=x^n$, then the derivative $f^{\prime }(x)=n{x}^{n-1}$.

4. Negative Exponent Rule: The negative exponent rule states that for any nonzero number $b$ and positive integer $n$, $b^{-n}=\frac{1}{b^n}$. This rule is used to simplify expressions with negative exponents.

5. Simplifying Expressions: The process of simplifying expressions involves combining like terms, reducing fractions, and applying mathematical rules to make the expression easier to understand or work with.

6. Derivative of a Function with Respect to x: When differentiating a function $y=f(x)$ with respect to $x$, we find $\frac{dy}{dx}$, which represents the rate of change of $y$ with respect to $x$.