Questions: Find the derivative of the function. A(r) = sqrt(r) * e^(r^4 + 3) A'(r) =

Solution Steps

To find the derivative of the function \( A(r) = \sqrt{r} \cdot e^{r^4 + 3} \), we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions \( u(r) \cdot v(r) \) is \( u'(r) \cdot v(r) + u(r) \cdot v'(r) \). Here, \( u(r) = \sqrt{r} \) and \( v(r) = e^{r^4 + 3} \). We will also need to apply the chain rule to find the derivative of \( v(r) \).

We start with the function defined as: \[ A(r) = \sqrt{r} \cdot e^{r^4 + 3} \]

To find the derivative \( A'(r) \), we apply the product rule: \[ A'(r) = u'(r) \cdot v(r) + u(r) \cdot v'(r) \] where \( u(r) = \sqrt{r} \) and \( v(r) = e^{r^4 + 3} \).

First, we calculate the derivatives of \( u(r) \) and \( v(r) \):

• The derivative of \( u(r) \): \[ u'(r) = \frac{1}{2\sqrt{r}} \]
• The derivative of \( v(r) \) using the chain rule: \[ v'(r) = e^{r^4 + 3} \cdot \frac{d}{dr}(r^4 + 3) = e^{r^4 + 3} \cdot 4r^3 \]

Substituting \( u(r) \), \( u'(r) \), \( v(r) \), and \( v'(r) \) back into the product rule gives: \[ A'(r) = \left(\frac{1}{2\sqrt{r}}\right) \cdot e^{r^4 + 3} + \sqrt{r} \cdot \left(4r^3 \cdot e^{r^4 + 3}\right) \]

Combining the terms, we have: \[ A'(r) = e^{r^4 + 3} \left(\frac{1}{2\sqrt{r}} + 4r^{7/2}\right) \]

Final Answer