Math Problem Statement

Find the derivative of the algebraic function. f(x) = 7x − 3/√x

Solution

To find the derivative of the given algebraic function f(x)=7x3xf(x) = 7x - \frac{3}{\sqrt{x}}, let's go step-by-step.

1. Rewrite the function

We'll rewrite 3x\frac{3}{\sqrt{x}} as a power of xx.

f(x)=7x3x12f(x) = 7x - 3x^{-\frac{1}{2}}

2. Differentiate term by term

Now, differentiate each term using basic differentiation rules:

  • The derivative of 7x7x is 77.
  • The derivative of 3x12-3x^{-\frac{1}{2}} is found using the power rule ddx[xn]=nxn1\frac{d}{dx} [x^n] = nx^{n-1}.

ddx(3x12)=3(12)x32=32x32\frac{d}{dx} \left( -3x^{-\frac{1}{2}} \right) = -3 \cdot \left( -\frac{1}{2} \right) x^{-\frac{3}{2}} = \frac{3}{2}x^{-\frac{3}{2}}

3. Combine the derivatives

Thus, the derivative of the function is:

f(x)=7+32x32f'(x) = 7 + \frac{3}{2}x^{-\frac{3}{2}}

Final answer:

f(x)=7+32x3f'(x) = 7 + \frac{3}{2\sqrt{x^3}}

Would you like more details on any part of this process?

Here are some related questions:

  1. What is the power rule for differentiation?
  2. How do you differentiate terms with fractional exponents?
  3. How do we simplify negative exponents in derivatives?
  4. How can derivatives be used to find the slope of a curve?
  5. What does the derivative tell us about the behavior of a function?

Tip: Always rewrite terms with roots or fractions into exponents when differentiating—it makes the process easier!

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Math Problem Analysis

Mathematical Concepts

Algebra
Calculus
Differentiation
Power Rule

Formulas

Derivative of x^n: d/dx[x^n] = nx^(n-1)
Derivative of a constant multiple: d/dx[c*f(x)] = c * d/dx[f(x)]

Theorems

Power Rule
Constant Multiple Rule

Suitable Grade Level

Grades 11-12 (or early college level)