Derivative Of Exponential Function: A Step-by-Step Guide

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Find the Derivative of the Exponential Function

Hey guys! Let's dive into finding the derivative of an exponential function. Specifically, we're going to tackle the function h(x) = 2(x2 - x). Don't worry, it's not as intimidating as it looks! We'll break it down step by step so you can easily understand how to find h'(x).

Understanding Exponential Functions and Derivatives

Before we jump into the problem, let's quickly recap what exponential functions and derivatives are all about. An exponential function is a function where the variable appears in the exponent. In our case, we have 2 raised to the power of (x^2 - x). Derivatives, on the other hand, tell us the rate at which a function is changing at any given point. Think of it as the slope of the tangent line to the function's graph.

When dealing with exponential functions, a crucial formula to remember is the derivative of a^x, where 'a' is a constant. The derivative of a^x with respect to x is a^x * ln(a), where ln(a) is the natural logarithm of 'a'. This formula forms the foundation for differentiating more complex exponential functions, like the one we're about to solve. It's also important to remember the chain rule, which we will use to find the derivative of the composite function in the exponent. So, keep these concepts in mind as we move forward. Understanding these basics is key to mastering the differentiation process and tackling more challenging problems down the road.

Applying the Chain Rule

Now, let's get to the fun part: finding the derivative of h(x) = 2(x2 - x). This requires using the chain rule, which is essential when dealing with composite functions. In simple terms, the chain rule helps us differentiate a function within a function. Here, our outer function is 2^u, where u = x^2 - x is the inner function. According to the chain rule, the derivative of h(x) will be the derivative of the outer function (2^u) with respect to u, multiplied by the derivative of the inner function (u = x^2 - x) with respect to x. This can be written as: h'(x) = (d/du)(2^u) * (d/dx)(x^2 - x).

So, the first step is to find the derivative of 2^u with respect to u. As we mentioned earlier, the derivative of a^x is a^x * ln(a). Applying this to our function, we get (d/du)(2^u) = 2^u * ln(2). Next, we need to find the derivative of the inner function, u = x^2 - x, with respect to x. Using the power rule, which states that the derivative of x^n is n*x^(n-1), we find that (d/dx)(x^2 - x) = 2x - 1. Now that we have both derivatives, we can multiply them together to find h'(x). Remember to substitute u = x^2 - x back into the equation. This step-by-step approach ensures that we correctly apply the chain rule, leading us to the final derivative. It's all about breaking down the problem into smaller, manageable parts and then putting them back together!

Calculating the Derivatives

Alright, let's put everything together. We know that:

  • (d/du)(2^u) = 2^u * ln(2)
  • (d/dx)(x^2 - x) = 2x - 1

Using the chain rule, we multiply these two derivatives:

h'(x) = (2^u * ln(2)) * (2x - 1)

Now, substitute u = x^2 - x back into the equation:

h'(x) = 2(x2 - x) * ln(2) * (2x - 1)

So, the derivative of h(x) = 2(x2 - x) is h'(x) = 2(x2 - x) * ln(2) * (2x - 1). And that's it! We've successfully found the derivative of our exponential function. Remember to take your time and double-check each step to avoid any common mistakes.

Simplifying the Expression

Although we've found the derivative, it's always good practice to see if we can simplify the expression further. In this case, h'(x) = 2(x2 - x) * ln(2) * (2x - 1) is already in a pretty simplified form. We can't really combine any of the terms or factor anything out. However, we can rewrite it slightly to make it look a bit cleaner:

h'(x) = (2x - 1) * ln(2) * 2(x2 - x)

This is just a matter of personal preference, but some people might find this arrangement more aesthetically pleasing. The important thing is that the expression is mathematically equivalent to our previous result. Sometimes, simplification might involve factoring, combining like terms, or using trigonometric identities. In our case, there aren't any obvious simplifications to make, so we can confidently say that our derivative is in its simplest form. Always remember that simplifying expressions can make them easier to work with and understand.

Common Mistakes to Avoid

When finding derivatives of exponential functions, there are a few common mistakes that you should watch out for. One of the most frequent errors is forgetting to apply the chain rule. Remember that if you have a composite function, you need to differentiate the outer function and then multiply by the derivative of the inner function. Another mistake is incorrectly applying the power rule to exponential functions. The power rule applies to terms like x^n, not to exponential terms like a^x. Also, be careful with the constant factors. Make sure you don't accidentally drop or misplace the ln(a) term when differentiating a^x.

To avoid these mistakes, it's helpful to practice a lot of problems and double-check your work. Pay close attention to the details and make sure you understand the underlying concepts. It's also a good idea to write out each step clearly, so you can easily spot any errors. By being mindful of these common pitfalls, you can improve your accuracy and confidence when differentiating exponential functions.

Practice Problems

To solidify your understanding, let's look at a couple of practice problems.

  1. Find the derivative of f(x) = 3(x3 + 2x).
  2. Find the derivative of g(x) = 5^(sin(x)).

Try solving these problems on your own, and then check your answers. The solutions are provided below:

Solution 1: f'(x) = 3(x3 + 2x) * ln(3) * (3x^2 + 2)

Solution 2: g'(x) = 5^(sin(x)) * ln(5) * cos(x)

Working through these practice problems will help you build your skills and gain confidence in differentiating exponential functions. The more you practice, the better you'll become at recognizing patterns and applying the appropriate techniques.

Conclusion

So, there you have it! We've successfully found the derivative of the exponential function h(x) = 2(x2 - x). Remember the key steps: understand exponential functions and derivatives, apply the chain rule, calculate the derivatives, and simplify the expression. By following these steps and avoiding common mistakes, you'll be well on your way to mastering the art of differentiating exponential functions. Keep practicing, and you'll become a pro in no time! Happy differentiating, folks!