https://file.aiquickdraw.com/mathbot/file/1727635062367eynxm0yw.jpg

We are given the function:

\[
f(x) = 7x^5\sqrt{x} + \frac{-4}{x^2\sqrt{x}}
\]

To find the derivative \( f'(x) \), we can simplify the terms and then apply the rules of differentiation.

### Step 1: Simplify each term

#### First term: \( 7x^5\sqrt{x} \)

Since \( \sqrt{x} = x^{1/2} \), the first term becomes:

\[
7x^5\sqrt{x} = 7x^5 \cdot x^{1/2} = 7x^{5 + 1/2} = 7x^{11/2}
\]

#### Second term: \( \frac{-4}{x^2\sqrt{x}} \)

We can rewrite the denominator as \( x^2 \cdot x^{1/2} = x^{5/2} \), so the second term becomes:

\[
\frac{-4}{x^{5/2}} = -4x^{-5/2}
\]

### Step 2: Differentiate each term

Now, we differentiate each term using the power rule \( \frac{d}{dx}x^n = n x^{n-1} \).

#### Derivative of the first term:

\[
\frac{d}{dx}\left(7x^{11/2}\right) = 7 \cdot \frac{11}{2}x^{11/2 - 1} = \frac{77}{2}x^{9/2}
\]

#### Derivative of the second term:

\[
\frac{d}{dx}\left(-4x^{-5/2}\right) = -4 \cdot \frac{-5}{2}x^{-5/2 - 1} = \frac{20}{2}x^{-7/2} = 10x^{-7/2}
\]

### Step 3: Combine the results

Thus, the derivative of the function is:

\[
f'(x) = \frac{77}{2}x^{9/2} + 10x^{-7/2}
\]

Would you like more details or have any questions?

Here are 5 related questions to further practice:

1. How would you differentiate \( f(x) = 5x^4\sqrt{x} + \frac{-6}{x^3\sqrt{x}} \)?
2. How can you rewrite a fractional power, such as \( \frac{1}{x^n} \), as a negative exponent?
3. What is the derivative of \( f(x) = x^{2/3} - 3x^{-1/2} \)?
4. How do you apply the chain rule for composite functions like \( \sqrt{2x + 1} \)?
5. What is the second derivative of the function \( f(x) = 7x^5\sqrt{x} + \frac{-4}{x^2\sqrt{x}} \)?

**Tip:** Always simplify terms with roots or fractional exponents before differentiating to make calculations easier.

Let f(x) = 7x^5√x + (-4) / (x^2√x). Find f'(x).

Learn how to differentiate the function f(x) = 7x^5√x + (-4)/(x^2√x) using the power rule. This problem covers key calculus concepts including simplifying expressions and applying differentiation rules. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation