Questions: Find the indicated derivative and simplify. dy/dx for y=25 x^(1/5)(x^5+6) dy/dx=

Solution Steps

To find the indicated derivative, we will use the product rule of differentiation. The product rule states that if you have a function \( y = u(x) \cdot v(x) \), then the derivative \( \frac{dy}{dx} \) is given by \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \). Here, \( u(x) = 25x^{\frac{1}{5}} \) and \( v(x) = x^5 + 6 \). We will find the derivatives \( u'(x) \) and \( v'(x) \) separately and then apply the product rule.

We start with the function \( y = 25 x^{\frac{1}{5}}(x^5 + 6) \). Here, we define:

• \( u = 25 x^{\frac{1}{5}} \)
• \( v = x^5 + 6 \)

Next, we compute the derivatives of \( u \) and \( v \):

• The derivative of \( u \) is given by: \[ u' = \frac{d}{dx}(25 x^{\frac{1}{5}}) = 5.0 x^{-\frac{4}{5}} = \frac{5.0}{x^{\frac{4}{5}}} \]
• The derivative of \( v \) is: \[ v' = \frac{d}{dx}(x^5 + 6) = 5 x^4 \]

Using the product rule \( \frac{dy}{dx} = u'v + uv' \), we find: \[ \frac{dy}{dx} = \left(5.0 \cdot \frac{1}{x^{\frac{4}{5}}}\right)(x^5 + 6) + (25 x^{\frac{1}{5}})(5 x^4) \] This simplifies to: \[ \frac{dy}{dx} = \frac{5.0(x^5 + 6)}{x^{\frac{4}{5}}} + 125 x^{\frac{21}{5}} \]

After simplification, we obtain: \[ \frac{dy}{dx} = \frac{130.0 x^5 + 30.0}{x^{\frac{4}{5}}} \]

Final Answer