Derivative Of H(x) = 9x³ - √x + 1/x²: Explained!

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Hey guys! Today, we're diving into the world of calculus to figure out the derivative of the function h(x) = 9x³ - √x + 1/x². This might seem a bit daunting at first, but don't worry, we'll break it down step by step. We’ll explore the fundamental concepts behind derivatives, apply the power rule, and tackle those pesky square roots and fractions. So, grab your thinking caps, and let’s get started!

Understanding Derivatives

Before we jump into the problem, let's quickly recap what a derivative actually is. In simple terms, the derivative of a function tells us the instantaneous rate of change of that function at any given point. Think of it as the slope of the tangent line to the curve at that point. Derivatives are super important in calculus and have tons of applications in physics, engineering, economics, and more.

To find the derivative, we use a set of rules, and the most common one we'll use today is the power rule. The power rule states that if you have a term in the form of axⁿ, where a is a constant and n is any real number, then the derivative of that term is nax^(n-1). Basically, you multiply by the exponent and then subtract 1 from the exponent. Got it? Great! Now, let’s apply this to our function.

Breaking Down the Function

Our function is h(x) = 9x³ - √x + 1/x². To make it easier to work with, let’s rewrite the terms with radicals and fractions using exponents. Remember that √x is the same as x^(1/2), and 1/x² is the same as x^(-2). So, we can rewrite our function as:

h(x) = 9x³ - x^(1/2) + x^(-2)

Now it looks much friendlier, right? We’ve transformed the function into a form where we can easily apply the power rule. Each term is now in the form axⁿ, which is exactly what we need. This step is crucial because it allows us to use the power rule directly without getting confused by the radicals and fractions. It’s all about making the problem as straightforward as possible. Trust me, rewriting terms like this will save you a lot of headaches in the long run.

Applying the Power Rule

Now comes the fun part! Let's apply the power rule to each term in our function. We’ll take it one term at a time to keep things organized and clear. This methodical approach will help prevent mistakes and ensure we get the correct derivative. Remember, the key is to multiply by the exponent and then subtract 1 from the exponent.

Term 1: 9x³

For the first term, 9x³, we have a = 9 and n = 3. Applying the power rule, we multiply 9 by 3 and then subtract 1 from the exponent:

Derivative = 3 * 9x^(3-1) = 27x²

So, the derivative of 9x³ is 27x². Easy peasy, right? This is a straightforward application of the power rule, and it sets the stage for the next terms.

Term 2: -x^(1/2)

For the second term, -x^(1/2), we have a = -1 and n = 1/2. Applying the power rule:

Derivative = (1/2) * (-1)x^((1/2)-1) = -(1/2)x^(-1/2)

The derivative of -x^(1/2) is -(1/2)x^(-1/2). Don't let the fractional exponent scare you; it's just a number, and the power rule still applies the same way. This term often trips people up, but you’ve got it!

Term 3: x^(-2)

For the third term, x^(-2), we have a = 1 and n = -2. Applying the power rule:

Derivative = -2 * 1x^(-2-1) = -2x^(-3)

So, the derivative of x^(-2) is -2x^(-3). Notice how the negative exponent comes into play here. This is another common area where mistakes can happen, so always double-check your signs!

Combining the Derivatives

Now that we've found the derivative of each term, we simply add them together to get the derivative of the entire function. Remember, the derivative of a sum (or difference) is the sum (or difference) of the derivatives.

So, the derivative of h(x) is:

h'(x) = 27x² - (1/2)x^(-1/2) - 2x^(-3)

We've done it! We've successfully found the derivative of h(x) by breaking it down into smaller, manageable parts and applying the power rule to each term. This is a great example of how calculus problems can be solved step by step. Now, let's match our result with the given options.

Matching the Answer

Looking at the options provided, our derivative h'(x) = 27x² - (1/2)x^(-1/2) - 2x^(-3) matches option A:

A) 27x² - (1/2)x^(-1/2) - 2/x³

Notice that -2x^(-3) is the same as -2/x³, so the expressions are equivalent. We’ve nailed it! Our step-by-step approach has led us to the correct answer. Always remember to double-check your work and match it against the given options to ensure accuracy.

Why Option A is Correct

To recap, we found the derivative of h(x) = 9x³ - √x + 1/x² by applying the power rule to each term after rewriting the function in a more manageable form. The power rule allowed us to easily find the derivative of each term, and then we combined these derivatives to get the derivative of the entire function.

  • The derivative of 9x³ is 27x².
  • The derivative of -√x (or -x^(1/2)) is -(1/2)x^(-1/2).
  • The derivative of 1/x² (or x^(-2)) is -2x^(-3), which is the same as -2/x³.

Adding these up, we get:

h'(x) = 27x² - (1/2)x^(-1/2) - 2/x³

This matches option A perfectly. The other options have incorrect signs or coefficients, making them incorrect. It’s crucial to pay attention to these details when working with derivatives.

Common Mistakes to Avoid

When finding derivatives, there are a few common mistakes that students often make. Let’s go over them so you can avoid these pitfalls:

  1. Forgetting the Power Rule: The power rule is the bread and butter of derivative calculations, especially for polynomial functions. Forgetting to multiply by the exponent or subtracting 1 from the exponent is a common mistake. Always double-check your application of the power rule.
  2. Incorrectly Rewriting Terms: Before applying the power rule, make sure to rewrite all radicals and fractions as exponents. For example, √x should be rewritten as x^(1/2), and 1/x² should be rewritten as x^(-2). Messing this up can lead to incorrect derivatives.
  3. Sign Errors: Pay close attention to the signs, especially when dealing with negative exponents and coefficients. A simple sign error can throw off the entire calculation.
  4. Not Distributing Correctly: If you have a function that involves sums or differences of terms, make sure to apply the derivative to each term individually. Don’t try to take shortcuts, or you might miss something.

By being aware of these common mistakes, you can avoid them and ensure that you get the correct derivative every time. Practice makes perfect, so keep working on these problems, and you’ll become a derivative master in no time!

Conclusion

So, there you have it! The derivative of h(x) = 9x³ - √x + 1/x² is 27x² - (1/2)x^(-1/2) - 2/x³, which corresponds to option A. We tackled this problem by understanding the definition of a derivative, applying the power rule, and breaking the function down into manageable parts. Remember, derivatives are a fundamental concept in calculus, and mastering them will open up a whole new world of mathematical possibilities.

I hope this explanation was helpful, guys! If you have any more questions or want to explore other calculus topics, just let me know. Keep practicing, and you’ll become a calculus whiz in no time. Happy calculating!