Math Problem Statement

Supply a valid technology formula for the given function. (7 + x)^6x x + 4

Solution

The function you provided looks like a composite expression, but it’s not clear if you're asking for simplification, differentiation, or some other operation involving technology formulas.

Given the expression (7+x)6x(7 + x)^{6x} and the second term x+4x + 4, I’ll provide some potential interpretations based on common requests:

1. Simplifying the expression:

There isn’t much that can be simplified directly between the terms (7+x)6x(7 + x)^{6x} and x+4x + 4. They are separate, and no simplifications can combine them.

2. Differentiating with respect to xx:

For differentiation, we can handle the two expressions separately:

  • For the term (7+x)6x(7 + x)^{6x}, we use the product rule combined with the chain rule and logarithmic differentiation.

Let y=(7+x)6xy = (7 + x)^{6x}. Take the natural logarithm of both sides to apply logarithmic differentiation:

lny=6xln(7+x)\ln y = 6x \ln(7 + x)

Differentiate both sides with respect to xx:

dydx1y=6ln(7+x)+6x17+x\frac{dy}{dx} \cdot \frac{1}{y} = 6 \ln(7 + x) + 6x \cdot \frac{1}{7 + x}

Multiply both sides by y=(7+x)6xy = (7 + x)^{6x} to solve for dydx\frac{dy}{dx}:

dydx=(7+x)6x(6ln(7+x)+6x7+x)\frac{dy}{dx} = (7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right)

  • For the second term x+4x + 4, its derivative is straightforward:

ddx(x+4)=1\frac{d}{dx} (x + 4) = 1

So the derivative of the entire function is:

ddx[(7+x)6x+(x+4)]=(7+x)6x(6ln(7+x)+6x7+x)+1\frac{d}{dx} \left[ (7 + x)^{6x} + (x + 4) \right] = (7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right) + 1

This would be the formula for the derivative of the function with respect to xx.

3. Using in technology (computational tools):

If you need to input this into a calculator or software like WolframAlpha or any symbolic computing software (like Python’s SymPy or MATLAB), you would write:

f(x) = (7 + x)^(6x) + (x + 4)
diff(f(x), x)

This tells the system to differentiate the function f(x)f(x) with respect to xx.

Summary:

The derivative of the function (7+x)6x+(x+4)(7 + x)^{6x} + (x + 4) is:

(7+x)6x(6ln(7+x)+6x7+x)+1(7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right) + 1

This is the most likely interpretation of your request.

Would you like more details or further clarifications?

5 Related Questions:

  1. How do you apply the product rule to functions with exponents involving xx?
  2. What is logarithmic differentiation, and when is it useful?
  3. Can you show how to solve the same function using a specific software like WolframAlpha or SymPy?
  4. How does the chain rule apply in this problem?
  5. What are the best strategies for simplifying complex exponential functions before differentiation?

Tip:

When dealing with complex exponentials, applying logarithmic differentiation early on simplifies many steps, especially when both the base and exponent depend on xx.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Product Rule
Logarithmic Differentiation
Chain Rule

Formulas

(7 + x)^{6x}
Derivative of a composite exponential function
Logarithmic Differentiation: ln(y) = 6x * ln(7 + x)

Theorems

Product Rule: d(uv)/dx = u'v + uv'
Chain Rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
Logarithmic Differentiation

Suitable Grade Level

Advanced High School (Grades 11-12) or Early University Level