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Chapter 4: Problem 9

Differentiate. $$ y=6^{x} $$

Short Answer

Expert verified
y = 6^{x} -> y = 6^{x} \ln(6)

Step by step solution

01

Understand the problem

The task is to differentiate the function given by the expression:
02

Recall the derivative rule for exponential functions

For any function of the form
03

Apply the derivative rule

Using the rule for differentiating exponential functions:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

calculus
Calculus is a branch of mathematics that deals with rates of change. It helps us understand how quantities change over time. It includes two main concepts: differentiation and integration. Differentiation studies how things change, while integration is about adding up those changes. Differentiation is like finding the slope of a curve, and integration involves finding the area under a curve. In this article, we focus on differentiation.
differentiation
Differentiation is the process of finding the derivative of a function. The derivative tells us how a function changes at any point. For example, the derivative of a position function can give us the velocity. To differentiate a function, we apply rules that help us find how quickly the function's value changes.

In this exercise, we differentiate the function: \( y=6^x \). This means we'll find the rate of change of \( y \) with respect to \( x \).
exponential functions
Exponential functions are functions where the variable is an exponent. An example is \( 6^x \). These functions grow or decay at rates proportional to their current value. This makes them common in nature, such as in populations and radioactive decay.

When differentiating exponential functions, we must be careful to use the correct rules to handle the exponent.
derivative rules
To differentiate exponential functions, we use specific derivative rules. For a function of the form \( a^x \), where \( a \) is a constant, the derivative is given by:
\[ \frac {d}{dx} \big( a^x \big) = a^x \times \text{ln}(a) \] In this formula, \( \text{ln}(a)\) is the natural logarithm of the base \( a \).

Applying this to our function, \( 6^x\), we get:
\[ y' =6^x \times \text{ln}(6) \]
mathematical problem solving
Mathematical problem solving involves applying appropriate methods and strategies to find a solution. Here are some steps to solve differentiation problems:
  • Understand the problem: Identify what needs to be found or proved.
  • Recall relevant rules or formulas: Use differentiation rules suitable for the given function.
  • Apply the rules: Perform calculations step-by-step, ensuring accuracy.
  • Verify: Check your solution for correctness and consistency.
By following these steps, we successfully differentiated \( 6^x \) to find its rate of change.

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Most popular questions from this chapter

Differentiate. $$ y=\log _{2} \sec x $$

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