Find dy/dx y=e^( cube root of x)

Solution:

Step 1:

Transform the cube root of $x$ into a fractional exponent: $\sqrt[3]{x} = x^{\frac{1}{3}}$. Thus, $y = e^{x^{\frac{1}{3}}}$.

Step 2:

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

Step 3:

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

Step 4:

Compute the derivative of the exponential function:

Step 4.1:

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

Step 4.1.1:

Set $u = x^{\frac{1}{3}}$ and differentiate: $\frac{d}{du} e^u \cdot \frac{d}{dx} x^{\frac{1}{3}}$.

Step 4.1.2:

Use the exponential rule: $\frac{d}{du} e^u = e^u \ln(e)$, thus $e^u \cdot \frac{d}{dx} x^{\frac{1}{3}}$.

Step 4.1.3:

Substitute $u$ back with $x^{\frac{1}{3}}$: $e^{x^{\frac{1}{3}}} \cdot \frac{d}{dx} x^{\frac{1}{3}}$.

Step 4.2:

Apply the power rule: $\frac{d}{dx} x^n = n x^{n-1}$, where $n = \frac{1}{3}$, to get $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{1}{3} - 1}$.

Step 4.3:

Express $-1$ as a fraction with a common denominator: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}$.

Step 4.4:

Combine the exponents: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{1}{3} + \frac{-3}{3}}$.

Step 4.5:

Simplify the exponent by adding the numerators over the common denominator: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{1 - 3}{3}}$.

Step 4.6:

Simplify the numerator:

Step 4.6.1:

Multiply $-1$ by $3$: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{1 - 3}{3}}$.

Step 4.6.2:

Subtract $3$ from $1$: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{\frac{-2}{3}}$.

Step 4.7:

Place the negative exponent in front of the fraction: $e^{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{-\frac{2}{3}}$.

Step 4.8:

Combine the constant and the power of $x$: $e^{x^{\frac{1}{3}}} \cdot \frac{x^{-\frac{2}{3}}}{3}$.

Step 4.9:

Merge the exponential function with the fraction: $\frac{e^{x^{\frac{1}{3}}} x^{-\frac{2}{3}}}{3}$.

Step 4.10:

Apply the negative exponent rule to move $x^{-\frac{2}{3}}$ to the denominator: $\frac{e^{x^{\frac{1}{3}}}}{3 x^{\frac{2}{3}}}$.

Step 5:

Formulate the equation by equating the left side to the right side: $y = \frac{e^{x^{\frac{1}{3}}}}{3 x^{\frac{2}{3}}}$.

Step 6:

Replace $y$ with $\frac{dy}{dx}$ to express the derivative: $\frac{dy}{dx} = \frac{e^{x^{\frac{1}{3}}}}{3 x^{\frac{2}{3}}}$.

Knowledge Notes:

1. Exponential Functions: An exponential function is of the form $f(x) = a^x$ where $a$ is a constant. The base $e$ is a special constant approximately equal to 2.71828, and it is the base of the natural logarithm.

2. Chain Rule: The chain rule is a formula for computing the derivative of the composition of two or more functions. In mathematical terms, if $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.

3. Power Rule: The power rule is used to differentiate functions of the form $f(x) = x^n$. The rule states that the derivative of such a function is $f'(x) = n x^{n-1}$.

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

5. Fractional Exponents: A fractional exponent represents a root. For example, $x^{\frac{1}{n}}$ is the $n$-th root of $x$. In the case of a cube root, $x^{\frac{1}{3}}$ is used.

6. Exponential Rule: The derivative of $e^u$ with respect to $u$ is $e^u \cdot \frac{du}{dx}$, where $u$ is a function of $x$. This is a specific case of the chain rule when the base of the exponential function is $e$.

By understanding these concepts, one can solve a variety of problems involving differentiation, especially when dealing with exponential functions and their compositions with other functions.