Hey calculus enthusiasts! Ever stared at a function and wondered, "How fast is this thing changing right now?" That, my friends, is the heart of calculus, and the derivative is your magic wand to find out. We're diving deep into examples of finding the derivative, so grab your thinking caps, and let's get this math party started!

What's a Derivative, Anyway?

Before we jump into the nitty-gritty of examples of finding the derivative, let's get a solid grip on what a derivative actually is. Think of it as the instantaneous rate of change of a function. If you're driving a car, your position is a function of time. The derivative of that function? That's your speedometer reading – how fast you're going at that exact moment. It tells you the slope of the tangent line to the function's graph at any given point. Super handy, right?

Now, there are a few ways to find these magical derivatives. We've got the definition, which involves limits (don't let that word scare you!), and then we have a bunch of awesome shortcut rules that make life so much easier. We'll explore both to give you the full picture.

The Foundational Way: Using the Limit Definition

Alright, guys, let's start with the OG method, the one that builds the foundation for everything else: the limit definition of the derivative. This is where we really understand why derivatives work. The formula looks a bit intimidating at first glance, but break it down, and it's totally manageable. The derivative of a function f(x), denoted as f'(x) (read as "f prime of x"), is defined as:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Let's unpack this beast. We're looking at the change in the function's value, f(x+h) - f(x), divided by the change in x, which is h. We're essentially finding the slope of the secant line between two points on the curve that are super close together. As h (the distance between those x-values) gets smaller and smaller, approaching zero, that secant line becomes the tangent line, and its slope is our derivative! Pretty neat, huh?

Example 1: Derivative of a Linear Function

Let's take a simple linear function: $f(x) = 3x + 2$. We want to find $f'(x)$ using the limit definition.

1. Find f(x+h): Replace every 'x' in $f(x)$ with '(x+h)'. $f(x+h) = 3(x+h) + 2 = 3x + 3h + 2$

2. Plug into the formula:

   $$f'(x) = \lim_{h \to 0} \frac{(3x + 3h + 2) - (3x + 2)}{h}$$

3. Simplify the numerator:

   $$f'(x) = \lim_{h \to 0} \frac{3x + 3h + 2 - 3x - 2}{h}$$

   $$f'(x) = \lim_{h \to 0} \frac{3h}{h}$$

4. Cancel out 'h' (since h is approaching 0, not equal to 0, we can cancel):

   $$f'(x) = \lim_{h \to 0} 3$$

5. Evaluate the limit: The limit of a constant is just the constant.

   $$f'(x) = 3$$

And there you have it! The derivative of $f(x) = 3x + 2$ is $f'(x) = 3$. This makes total sense because a linear function has a constant slope of 3 everywhere. See? The limit definition works!

Example 2: Derivative of a Quadratic Function

Now, let's amp it up a bit with a quadratic function: $f(x) = x^2$. Finding its derivative using limits will show us how the rate of change can vary.

1. Find f(x+h): $f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$

2. Plug into the formula:

   $$f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h}$$

3. Simplify the numerator:

   $$f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$$

   $$f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}$$

4. Factor out 'h' from the numerator:

   $$f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}$$

5. Cancel out 'h':

   $$f'(x) = \lim_{h \to 0} (2x + h)$$

6. Evaluate the limit: As h approaches 0, the term 'h' disappears.

   $$f'(x) = 2x$$

Boom! The derivative of $f(x) = x^2$ is $f'(x) = 2x$. This means the slope of the tangent line to the parabola $y=x^2$ changes depending on the value of x. At x=1, the slope is 2(1)=2. At x=3, the slope is 2(3)=6. It's increasing as x increases, just like the graph suggests!

While the limit definition is fundamental for understanding, it can get pretty lengthy for more complex functions. That's where the power rules come in!

Shortcut Superstars: The Power Rule and Beyond

So, you've seen the limit definition. It's the bedrock, the pure truth of derivatives. But honestly, nobody wants to do all that algebraic wrestling every single time, especially when you're dealing with functions like $f(x) = 5x^7 - 2x^3 + 10x$. That's where calculus gods gifted us with differentiation rules, and the Power Rule is the MVP.

The Power Rule: Your New Best Friend

If you have a function in the form $f(x) = ax^n$, where 'a' is a constant and 'n' is any real number, the Power Rule says its derivative is:

$$f'(x) = a n x^{n-1}$$

See what happened there? You take the exponent ('n'), bring it down to the front as a multiplier, and then subtract 1 from the original exponent. Easy peasy!

Example 3: Applying the Power Rule

Let's revisit $f(x) = x^2$ using the Power Rule.

• Here, $a=1$ and $n=2$.
• Bring the exponent (2) down: $2 n x^n$
• Subtract 1 from the exponent: $2-1 = 1$
• So, $f'(x) = 1 n 2 n x^{2-1} = 2x^1 = 2x$.

See? We got the same answer as with the limit definition, but in a fraction of the time! This is why the Power Rule is so incredibly useful for examples of finding the derivative.

Example 4: More Power Rule Fun

Let's try $g(x) = 5x^3$.

• $a=5$, $n=3$.
• Bring down the 3: $5 n 3 n x^n$
• Subtract 1 from the exponent: $3-1 = 2$
• $g'(x) = 5 n 3 n x^{3-1} = 15x^2$.

Example 5: What About Constants?

What's the derivative of a constant, like $f(x) = 7$? Think about the graph of $y=7$. It's a flat horizontal line. Does it have a slope? Nope, its slope is zero everywhere. Using the Power Rule, we can write $f(x) = 7x^0$ (since $x^0 = 1$).

• $a=7$, $n=0$.
• Bring down the 0: $7 n 0 n x^n$
• Subtract 1 from the exponent: $0-1 = -1$
• $f'(x) = 7 n 0 n x^{-1} = 0$.

The derivative of any constant is always 0. This is a super important rule to remember!

The Constant Multiple Rule

This rule is closely related to the Power Rule. If you have a constant multiplied by a function, like $f(x) = c n g(x)$, then its derivative is $f'(x) = c n g'(x)$. You just keep the constant and differentiate the function part.

Example 6: Constant Multiple Rule in Action

Let's find the derivative of $h(x) = -4x^5$.

• The constant is -4, and the function part is $x^5$.
• We know the derivative of $x^5$ using the Power Rule is $5x^{5-1} = 5x^4$.
• So, $h'(x) = -4 n (5x^4) = -20x^4$.

Piece of cake, right?

The Sum and Difference Rule

This rule is a lifesaver when you have functions added or subtracted together. If you have $f(x) = g(x) n ext{or} n h(x)$, then its derivative is $f'(x) = g'(x) n ext{or} n h'(x)$. You just find the derivative of each part separately and keep the plus or minus sign.

Example 7: Combining Rules!

Let's find the derivative of $f(x) = 5x^3 - 2x^2 + 7x - 10$.

• Derivative of 5x³: Using the Power Rule and Constant Multiple Rule, we get $5 n 3x^{3-1} = 15x^2$.
• Derivative of -2x²: Using the Power Rule and Constant Multiple Rule, we get $-2 n 2x^{2-1} = -4x^1 = -4x$.
• Derivative of 7x: This is $7x^1$. So, $7 n 1x^{1-1} = 7x^0 = 7 n 1 = 7$.
• Derivative of -10: This is a constant, so its derivative is 0.

Now, combine them with the signs: $f'(x) = 15x^2 - 4x + 7 - 0$ $f'(x) = 15x^2 - 4x + 7$

And that's how you differentiate polynomials like a pro!

Derivatives of Other Common Functions

Calculus isn't just about polynomials, though. We often encounter trigonometric, exponential, and logarithmic functions. Thankfully, their derivatives are also well-defined and follow specific rules.

Trigonometric Functions

These are super common in physics and engineering. You'll want to memorize these:

• If $f(x) = \sin(x)$, then $f'(x) = \cos(x)$
• If $f(x) = \cos(x)$, then $f'(x) = -\sin(x)$
• If $f(x) = \tan(x)$, then $f'(x) = \sec^2(x)$

Example 8: Differentiating Trig Functions

Let's find the derivative of $f(x) = 3\cos(x) + \tan(x)$.

• Derivative of $3\cos(x)$ is $3 n (-\sin(x)) = -3\sin(x)$.
• Derivative of $\tan(x)$ is $\sec^2(x)$.
• So, $f'(x) = -3\sin(x) + \sec^2(x)$.

Exponential and Logarithmic Functions

The exponential function $e^x$ is special because its derivative is itself!

• If $f(x) = e^x$, then $f'(x) = e^x$

For a general exponential function $a^x$:

• If $f(x) = a^x$, then $f'(x) = a^x \ln(a)$

And for the natural logarithm:

• If $f(x) = \ln(x)$, then $f'(x) = \frac{1}{x}$

Example 9: Exponential Derivative

Find the derivative of $g(x) = 5^x + e^x$.

• Derivative of $5^x$ is $5^x \ln(5)$.
• Derivative of $e^x$ is $e^x$.
• So, $g'(x) = 5^x \ln(5) + e^x$.

Product Rule and Quotient Rule

What if your function is a product of two functions, like $f(x) = (x^2)(\sin x)$? Or a quotient, like $g(x) = \frac{e^x}{x}$? For these, we need the Product Rule and the Quotient Rule.

• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. (Derivative of the first times the second, plus the first times the derivative of the second).
• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$. (Derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared).

Example 10: Product Rule

Let's find the derivative of $f(x) = x^3 \sin(x)$.

• Let $u(x) = x^3$, so $u'(x) = 3x^2$.
• Let $v(x) = \sin(x)$, so $v'(x) = \cos(x)$.
• Using the Product Rule: $f'(x) = (3x^2)(\sin x) + (x^3)(\cos x)$.
• $f'(x) = 3x^2 \sin x + x^3 \cos x$.

These rules might seem like a lot, but with practice, they become second nature. The key is to identify the structure of your function and apply the correct rule.

Chain Rule: The Ultimate Game Changer

Now, let's talk about the Chain Rule. This is arguably one of the most powerful and frequently used rules in differentiation. It's for when you have a function inside another function – a composite function. Think of it like Russian nesting dolls.

If $y = f(g(x))$, then the derivative $y'$ is found by differentiating the outer function with respect to the inner function, and then multiplying by the derivative of the inner function. In notation:

$$y' = f'(g(x)) n g'(x)$$

Example 11: Basic Chain Rule

Find the derivative of $f(x) = (x^2 + 1)^3$.

• Here, the outer function is $(\cdot)^3$ and the inner function is $g(x) = x^2 + 1$.
• Derivative of the outer function (treating the inside as a variable): $3(\cdot)^2$. Evaluate this at the inner function: $3(x^2 + 1)^2$.
• Derivative of the inner function $g(x) = x^2 + 1$ is $g'(x) = 2x$.
• Multiply them together: $f'(x) = 3(x^2 + 1)^2 n (2x)$.
• Simplify: $f'(x) = 6x(x^2 + 1)^2$.

Example 12: Chain Rule with Trig

Find the derivative of $h(x) = \sin(4x)$.

• Outer function: $\sin(\cdot)$. Its derivative is $\cos(\cdot)$. Evaluated at the inner function: $\cos(4x)$.
• Inner function: $g(x) = 4x$. Its derivative is $g'(x) = 4$.
• Multiply: $h'(x) = \cos(4x) n 4 = 4\cos(4x)$.

Example 13: Chain Rule with Exponential

Find the derivative of $k(x) = e^{3x^2}$.

• Outer function: $e^{(\cdot)}$. Its derivative is $e^{(\cdot)}$. Evaluated at the inner function: $e^{3x^2}$.
• Inner function: $g(x) = 3x^2$. Its derivative is $g'(x) = 6x$.
• Multiply: $k'(x) = e^{3x^2} n (6x) = 6xe^{3x^2}$.

The Chain Rule is essential. You'll see it used in conjunction with the Product and Quotient rules too, making for some complex-looking but manageable derivatives!

Putting It All Together: More Practice Examples

Let's wrap up with a few more challenging examples of finding the derivative that combine several rules. This is where you really solidify your understanding.

Example 14: Mixed Rules

Find the derivative of $f(x) = \frac{\sin(x^2)}{e^{5x}}$.

This requires the Quotient Rule and the Chain Rule.

• Let $u(x) = \sin(x^2)$ and $v(x) = e^{5x}$.
• Find u'(x): Using the Chain Rule on $\sin(x^2)$. The derivative of $\sin(\cdot)$ is $\cos(\cdot)$. The derivative of $x^2$ is $2x$. So, $u'(x) = \cos(x^2) n 2x = 2x\cos(x^2)$.
• Find v'(x): Using the Chain Rule on $e^{5x}$. The derivative of $e^{(\cdot)}$ is $e^{(\cdot)}$. The derivative of $5x$ is $5$. So, $v'(x) = e^{5x} n 5 = 5e^{5x}$.
• Apply the Quotient Rule: $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$

  $$f'(x) = \frac{(2x\cos(x^2))(e^{5x}) - (\sin(x^2))(5e^{5x})}{(e^{5x})^2}$$

• Simplify (optional but good practice): Factor out $e^{5x}$ from the numerator.

  $$f'(x) = \frac{e^{5x}(2x\cos(x^2) - 5\sin(x^2))}{e^{10x}}$$

  $$f'(x) = \frac{2x\cos(x^2) - 5\sin(x^2)}{e^{5x}}$$

Example 15: Logarithmic Differentiation (A Sneaky Trick!)

Sometimes, differentiating functions involving products, quotients, and powers of functions can be simplified using logarithms. Let's find the derivative of $y = x^x$. This isn't a simple power rule or exponential rule!

1. Take the natural logarithm of both sides: $\ln(y) = \ln(x^x)$

2. Use logarithm properties to simplify: $\ln(y) = x \ln(x)$

3. Differentiate both sides implicitly with respect to x: Remember the Chain Rule on the left side ($ \frac{1}{y} n y' $) and the Product Rule on the right side.

   $$\frac{1}{y} y' = (1) \ln(x) + x \left( \frac{1}{x} \right)$$

   $$\frac{1}{y} y' = \ln(x) + 1$$

4. Solve for y':

   $$y' = y (\ln(x) + 1)$$

5. Substitute back the original expression for y: Since $y = x^x$.

   $$y' = x^x (\ln(x) + 1)$$

This technique is super useful for tricky functions!

Conclusion: You've Got This!

We've journeyed through the fascinating world of examples of finding the derivative, from the fundamental limit definition to the super-efficient shortcut rules like the Power Rule, Product Rule, Quotient Rule, and the all-powerful Chain Rule. Remember, practice is absolutely key. The more functions you differentiate, the more comfortable you'll become with recognizing patterns and applying the right rules. Keep working through problems, and don't be afraid to revisit the basics when needed. Happy differentiating, calculus crew!