Practice: Calculating Simple Derivatives

Okay, let's put the rules we've learned into practice. Calculating derivatives might seem abstract at first, but like learning any new skill, practice makes it much more concrete. We'll use the rules we just covered: the Constant Rule, the Power Rule, and the Sum Rule.

Remember the core ideas:

• The derivative tells us the instantaneous rate of change, or the slope of the function's graph at any given point.
• Constant Rule: The derivative of a constant $c$ is 0. If $f(x) = c$, then $f'(x) = 0$.
• Power Rule: The derivative of $x^n$ is $nx^{n-1}$. If $f(x) = ax^n$, then $f'(x) = a \cdot nx^{n-1}$.
• Sum Rule: The derivative of a sum of terms is the sum of their derivatives. If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.

Let's work through some examples.

Example 1: A Constant Function

Find the derivative of the function $f(x) = 15$.

• Identify the Rule: This function is a constant value. It doesn't depend on $x$. So, we use the Constant Rule.
• Apply the Rule: The derivative of any constant is 0.
• Result: $$f'(x) = 0$$
• Interpretation: The graph of $y = 15$ is a horizontal line. Its slope is always 0, meaning its value doesn't change as $x$ changes.

Example 2: A Simple Power Function

Let $y = x^5$. Find the derivative $\frac{dy}{dx}$.

• Identify the Rule: This is of the form $x^n$, where $n=5$. We use the Power Rule.
• Apply the Rule: Bring the exponent ($n=5$) down as a multiplier and reduce the exponent by 1 ($n-1 = 5-1 = 4$).
• Result: $$\frac{dy}{dx} = 5x^{5-1} = 5x^4$$
• Interpretation: The slope of the graph of $y = x^5$ is not constant. It depends on the value of $x$. For example, at $x=1$, the slope is $5(1)^4 = 5$. At $x=2$, the slope is $5(2)^4 = 5 \times 16 = 80$.

Example 3: A Power Function with a Coefficient

Find the derivative of $g(x) = 4x^3$.

• Identify the Rule: This is a constant ($4$) multiplied by a power function ($x^3$). We use the Power Rule combined with the fact that constants "pass through" the differentiation process.
• Apply the Rule: Keep the constant $4$. Apply the power rule to $x^3$ (bring down the 3, reduce the exponent to 2). Then multiply.
• Result: $$g'(x) = 4 \cdot (3x^{3-1}) = 4 \cdot (3x^2) = 12x^2$$

Example 4: Combining Rules - A Polynomial

Let's find the derivative of a polynomial function: $h(x) = 2x^3 + 7x^2 - 5x + 1$.

• Identify the Rules: This function is a sum of terms. We use the Sum Rule to differentiate each term separately. Each term involves constants and powers of $x$, so we'll also use the Constant Rule and Power Rule.
• Apply the Rules (term by term):
  • Derivative of $2x^3$: Apply the power rule. $2 \cdot (3x^{3-1}) = 6x^2$.
  • Derivative of $7x^2$: Apply the power rule. $7 \cdot (2x^{2-1}) = 14x^1 = 14x$.
  • Derivative of $-5x$: Remember $x$ is $x^1$. Apply the power rule. $-5 \cdot (1x^{1-1}) = -5 \cdot (1x^0) = -5 \cdot (1) = -5$. (Recall $x^0 = 1$).
  • Derivative of $1$: This is a constant. Apply the constant rule. The derivative is $0$.
• Combine the Results (using the Sum Rule): Add the derivatives of each term together.
• Result: $$h'(x) = 6x^2 + 14x - 5 + 0 = 6x^2 + 14x - 5$$

Example 5: Function with a Fractional Exponent

Find the derivative of $f(x) = 8\sqrt{x}$.

• Rewrite the Function: First, it's helpful to rewrite the square root using a fractional exponent. Remember that $\sqrt{x} = x^{1/2}$. So, $f(x) = 8x^{1/2}$.
• Identify the Rule: Now it looks like a constant times a power function. We use the Power Rule with $n = 1/2$.
• Apply the Rule: Keep the constant $8$. Apply the power rule to $x^{1/2}$. Bring down the exponent $1/2$ and subtract 1 from the exponent ($1/2 - 1 = -1/2$).
• Result: $$f'(x) = 8 \cdot (\frac{1}{2} x^{1/2 - 1}) = 8 \cdot (\frac{1}{2} x^{-1/2}) = 4x^{-1/2}$$
• Optional Simplification: You might also write this using radicals: $4x^{-1/2} = \frac{4}{x^{1/2}} = \frac{4}{\sqrt{x}}$. Both $4x^{-1/2}$ and $\frac{4}{\sqrt{x}}$ are correct representations of the derivative.

Try It Yourself

Now it's your turn. Use the rules we've discussed to find the derivatives of the following functions. Don't worry if it takes a few tries, the goal is to build familiarity.

1. $f(x) = -50$
2. $y = x^{10}$
3. $g(t) = 6t^4 - 3t^2 + 9t$ (Using $t$ instead of $x$ changes nothing about the rules!)
4. $h(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 + x$
5. $f(z) = z^2 - 7$

Take your time and work through each term. Check if you are applying the Constant Rule, Power Rule, and Sum Rule correctly. Calculating derivatives is a fundamental operation, and getting comfortable with these basic rules will be very helpful as we move towards optimization.