What is the general power rule for differentiating $f(x) = x^n$?

The derivative is $f'(x) = nx^{n-1}$. You multiply the term by the original power $n$ and then decrease the power by 1.

Why is the derivative of a constant (e.g., $y = 5$) always zero?

A constant function represents a horizontal line on a graph. Since the gradient (slope) of a horizontal line is zero at every point, its derivative is zero.

How should you prepare the term $\frac{3}{x^2}$ before differentiating?

You should rewrite it in index form as $3x^{-2}$. This allows you to apply the power rule directly to the exponent $-2$.

What is the difference between the derivative of $5x$ and the derivative of $x^5$?

The derivative of $5x$ is the constant $5$ (linear gradient), while the derivative of $x^5$ is $5x^4$ (a new function representing a changing gradient).

When differentiating $x^{-3}$, what is a common mistake regarding the new power?

A common error is writing the new power as $-2$. Because you must subtract 1, the correct new power is $-3 - 1 = -4$.

How do you differentiate an expression with multiple terms, like $x^3 + 4x^2 - 5$?

You use the sum rule, differentiating each term independently to get $3x^2 + 8x + 0$.

What does the notation $\frac{dy}{dx}$ specifically represent?

It represents the 'infinitesimal' change in $y$ divided by the 'infinitesimal' change in $x$, which is the definition of the gradient at a point.

How do you convert $\sqrt[3]{x}$ into a form ready for differentiation?

You rewrite the cube root as a fractional power: $x^{1/3}$. Then you can apply the power rule to get $\frac{1}{3}x^{-2/3}$.

What is the result of differentiating $y = x$?

The derivative is $1$. Since $x$ is $x^1$, the power rule gives $1x^0$, and any non-zero number to the power of 0 is 1.

If a curve is 'smooth', what does that imply about its derivative?

It implies that a unique tangent exists at every point, meaning the derivative is defined and continuous across that interval.

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Differentiation

Differentiating Powers of x

Summary

Differentiation is a fundamental calculus operation used to determine the instantaneous rate of change of a function. For power functions of the form $x^n$ , the power rule provides a systematic method to find the derivative, which geometrically represents the gradient of the tangent to the curve at any given point.

1. Definition & Core Concepts

A graph showing a curve with a tangent line at a specific point, illustrating that the derivative represents the slope of that tangent.

2. The Power Rule Methodology

3. Special Cases: Constants and Linear Terms

4. Preparation: Rewriting Expressions

5. Key Distinctions

6. Exam Strategy & Common Pitfalls

Negative Exponents: When subtracting 1 from a negative power, the number becomes 'more negative' (e.g., $-2 - 1 = -3$ ). Students often mistakenly move toward zero.
Fractional Arithmetic: Always double-check subtractions involving fractions, such as $\frac{1}{2} - 1 = -\frac{1}{2}$ .
Coefficient Handling: Ensure the original coefficient is multiplied by the entire original power, including its sign.
Final Form: Unless specified, it is often helpful to convert negative or fractional powers back into roots or fractions to verify the logic of the answer.