Math Problem Statement

احسب مشتقة الدالة g

احسب مشتقة الدالة g

Solution

The given function is:

g(x)=5x12xg(x) = \sqrt{5x} - \frac{1}{2x}

To find the derivative g(x)g'(x), we will use the following rules of differentiation:

  1. The derivative of x=x1/2\sqrt{x} = x^{1/2} is given by 12x\frac{1}{2\sqrt{x}}.
  2. The derivative of 1x\frac{1}{x} is 1x2-\frac{1}{x^2}.
  3. The power rule for derivatives, ddx[xn]=nxn1\frac{d}{dx}[x^n] = n x^{n-1}.

Now, let's compute the derivative of each term.

Step 1: Derivative of 5x\sqrt{5x}

We can rewrite 5x\sqrt{5x} as (5x)1/2(5x)^{1/2}. By applying the chain rule: ddx((5x)1/2)=12(5x)1/25=525x\frac{d}{dx}\left( (5x)^{1/2} \right) = \frac{1}{2}(5x)^{-1/2} \cdot 5 = \frac{5}{2\sqrt{5x}}

Step 2: Derivative of 12x-\frac{1}{2x}

This is 121x-\frac{1}{2} \cdot \frac{1}{x}. Using the rule for 1x\frac{1}{x}, we get: ddx(12x)=12x2\frac{d}{dx}\left( -\frac{1}{2x} \right) = \frac{1}{2x^2}

Final derivative g(x)g'(x)

Putting it all together: g(x)=525x+12x2g'(x) = \frac{5}{2\sqrt{5x}} + \frac{1}{2x^2}

This is the derivative of the given function.

Would you like any further explanation or details?

Related Questions:

  1. How can the chain rule be applied to other functions?
  2. What is the second derivative of this function g(x)g(x)?
  3. How would you compute the derivative of a product of two functions?
  4. How do you differentiate a function with multiple variables?
  5. Can you explain the process of implicit differentiation?

Tip:

Always simplify functions as much as possible before differentiating to avoid unnecessary complexity.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule
Power Rule

Formulas

Derivative of sqrt(x): d/dx(sqrt(x)) = 1/(2sqrt(x))
Derivative of 1/x: d/dx(1/x) = -1/x^2
Power Rule: d/dx[x^n] = n*x^(n-1)

Theorems

Chain Rule
Power Rule

Suitable Grade Level

University Level