Differentiate.
Upper G left parenthesis x right parenthesisequals​(3xsquaredplus5​)left parenthesis 2 x plus StartRoot x EndRoot right parenthesis


To differentiate the function \( G(x) = (3x^2 + 5)(2x + \sqrt{x}) \), we will apply the **product rule** because this is the product of two functions, \( u(x) = 3x^2 + 5 \) and \( v(x) = 2x + \sqrt{x} \).

The product rule states:

\[
\frac{d}{dx}[u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x)
\]

Let's differentiate step by step:

1. Differentiate \( u(x) = 3x^2 + 5 \):
   \[
   u'(x) = 6x
   \]

2. Differentiate \( v(x) = 2x + \sqrt{x} \):
   \[
   v'(x) = 2 + \frac{1}{2\sqrt{x}}
   \]

Now apply the product rule:

\[
G'(x) = u'(x)v(x) + u(x)v'(x)
\]

Substituting the values of \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \):

\[
G'(x) = (6x)(2x + \sqrt{x}) + (3x^2 + 5)\left(2 + \frac{1}{2\sqrt{x}}\right)
\]

Simplify each term:

\[
G'(x) = 6x(2x + \sqrt{x}) + (3x^2 + 5)\left(2 + \frac{1}{2\sqrt{x}}\right)
\]
\[
G'(x) = 12x^2 + 6x\sqrt{x} + (3x^2 + 5)\left(2 + \frac{1}{2\sqrt{x}}\right)
\]
Expand the second term:
\[
G'(x) = 12x^2 + 6x\sqrt{x} + (3x^2)(2) + (3x^2)\left(\frac{1}{2\sqrt{x}}\right) + 5(2) + 5\left(\frac{1}{2\sqrt{x}}\right)
\]
\[
G'(x) = 12x^2 + 6x\sqrt{x} + 6x^2 + \frac{3x^{3/2}}{2} + 10 + \frac{5}{2\sqrt{x}}
\]

Now, combine like terms to get the final result:

\[
G'(x) = 18x^2 + 6x\sqrt{x} + \frac{3x^{3/2}}{2} + 10 + \frac{5}{2\sqrt{x}}
\]

This is the derivative of \( G(x) \).

Would you like further clarification or detailed steps?

Here are 5 related questions for further practice:
1. What is the derivative of \( (x^2 + 1)(x + \sqrt{x}) \)?
2. Differentiate \( (x^3 + 2x)(3x^2 + 4x) \) using the product rule.
3. Apply the chain rule to differentiate \( \sqrt{x^2 + 5x} \).
4. Find the second derivative of \( G(x) = (3x^2 + 5)(2x + \sqrt{x}) \).
5. Can you differentiate \( (x + 1)(x^2 + \sqrt{x}) \)?

**Tip:** Remember that the product rule is useful whenever two functions are multiplied. Always ensure each part is differentiated carefully before simplifying.

Differentiate G(x) = (3x^2 + 5)(2x + √x)

Learn how to differentiate the function G(x) = (3x^2 + 5)(2x + √x) step by step using the product rule. This solution covers key differentiation concepts, including the product rule, power rule, and algebraic simplifications. Ideal for students in Grades 11-12 or Calculus 1.

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