Math Problem Statement

Find the derivative of the function f(x) = (4 + x)^6 + x, x > -4. Express f'(x) in terms of x.

Solution

We are given the function:

f(x)=(4+x)6+xf(x) = (4 + x)^6 + x

We are asked to find the derivative f(x)f'(x).

Step-by-step solution:

  1. Derivative of (4+x)6(4 + x)^6: This is a composite function, so we will use the chain rule.

    The chain rule states that the derivative of (g(x))n(g(x))^n is:

    ddx[(g(x))n]=n(g(x))n1g(x)\frac{d}{dx}[(g(x))^n] = n \cdot (g(x))^{n-1} \cdot g'(x)

    For (4+x)6(4 + x)^6, we have:

    • g(x)=4+xg(x) = 4 + x, so g(x)=1g'(x) = 1
    • n=6n = 6

    Applying the chain rule:

    ddx[(4+x)6]=6(4+x)51=6(4+x)5\frac{d}{dx}[(4 + x)^6] = 6 \cdot (4 + x)^5 \cdot 1 = 6(4 + x)^5

  2. Derivative of xx:

    The derivative of xx is simply 1.

  3. Combine the results:

    Now, sum the derivatives:

    f(x)=6(4+x)5+1f'(x) = 6(4 + x)^5 + 1

Final answer:

f(x)=6(4+x)5+1f'(x) = 6(4 + x)^5 + 1


Would you like more details on any of the steps? Here are some related questions for practice:

  1. What is the derivative of (x2+1)7(x^2 + 1)^7?
  2. How would you apply the chain rule to sin(3x)\sin(3x)?
  3. What is the derivative of (2x+5)4(2x + 5)^4?
  4. How does the chain rule apply to functions like ln(4x+1)\ln(4x + 1)?
  5. Find the second derivative of f(x)=(4+x)6+xf(x) = (4 + x)^6 + x.

Tip: When using the chain rule, always differentiate the "inner" and "outer" functions separately and then multiply them together.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Chain Rule

Formulas

Chain Rule: d/dx [g(x)^n] = n * g(x)^(n-1) * g'(x)

Theorems

Chain Rule

Suitable Grade Level

Grades 10-12