Find dy/dx y=x^3e^(x^5)

Solution:

Problem Rephrasing:

Calculate the derivative of the function $y = x^3 e^{x^5}$ with respect to $x$.

Solution Process:

Step 1:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(x^3 e^{x^5})$.

Step 2:

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

Step 3:

Proceed to differentiate the expression on the right-hand side.

Step 3.1:

Apply the Product Rule for differentiation: $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$, where $u = x^3$ and $v = e^{x^5}$.

Step 3.2:

Invoke the Chain Rule for differentiation: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(u) = e^u$ and $g(x) = x^5$.

Step 3.2.1:

Set $u = x^5$ to apply the Chain Rule: $x^3 \left( \frac{d}{du}(e^u) \frac{dx}{dx}(x^5) \right) + e^{x^5} \frac{dx}{dx}(x^3)$.

Step 3.2.2:

Differentiate using the rule for exponentials: $\frac{d}{du}(e^u) = e^u \ln(e)$, where $e$ is the base of the natural logarithm.

Step 3.2.3:

Substitute $x^5$ back in for $u$: $x^3 (e^{x^5} \frac{dx}{dx}(x^5)) + e^{x^5} \frac{dx}{dx}(x^3)$.

Step 3.3:

Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 5$.

Step 3.4:

Combine the multiplication of $x^3$ and $x^4$ by adding their exponents.

Step 3.4.1:

Rearrange the terms: $x^4 x^3 (e^{x^5} \cdot 5) + e^{x^5} \frac{dx}{dx}(x^3)$.

Step 3.4.2:

Use the exponent combination rule: $a^m a^n = a^{m+n}$.

Step 3.4.3:

Add the exponents: $4$ and $3$ to get $x^7 (e^{x^5} \cdot 5) + e^{x^5} \frac{dx}{dx}(x^3)$.

Step 3.5:

Position the constant $5$ in front of $e^{x^5}$.

Step 3.6:

Differentiate $x^3$ using the Power Rule with $n = 3$.

Step 3.7:

Simplify the expression.

Step 3.7.1:

Reorder the terms to $5 e^{x^5} x^7 + 3 e^{x^5} x^2$.

Step 3.7.2:

Rearrange the factors to $5 x^7 e^{x^5} + 3 x^2 e^{x^5}$.

Step 4:

Combine the terms to form the final derivative expression: $\frac{dy}{dx} = 5 x^7 e^{x^5} + 3 x^2 e^{x^5}$.

Step 5:

Replace $y$ with $\frac{dy}{dx}$ to complete the differentiation: $\frac{dy}{dx} = 5 x^7 e^{x^5} + 3 x^2 e^{x^5}$.

Knowledge Notes:

The problem involves finding the derivative of a function that is the product of two separate functions, $x^3$ and $e^{x^5}$. To solve this, one must be familiar with several rules of differentiation:

1. Product Rule: This rule is used when differentiating a product of two functions, which states that the derivative of $u(x)v(x)$ is $u'(x)v(x) + u(x)v'(x)$.

2. Chain Rule: This rule is applied when differentiating a composite function. It states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$.

3. Power Rule: This rule is used to differentiate functions of the form $x^n$, where the derivative is $nx^{n-1}$.

4. Exponential Rule: When differentiating the exponential function $e^u$, where $u$ is a function of $x$, the derivative is $e^u u'$.

5. Combining Exponents: When multiplying like bases, the exponents are added together, as in $a^m a^n = a^{m+n}$.

In this problem, the Product Rule is initially applied, followed by the Chain Rule for the exponential part, and the Power Rule for the polynomial part. The process involves setting up intermediate variables, differentiating, and then substituting back to simplify the expression to its final form.