First Derivative

First Derivative

Same as the derivative. We say first derivative instead of just derivative whenever there may be confusion between the first derivative and the second derivative (or the nth derivative).

Key Formula

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

Where:

$f(x)$ = The original function
$f'(x)$ = The first derivative of f with respect to x
$h$ = A small increment that approaches zero

Worked Example

Problem: Find the first derivative of f(x) = 3x⁴ − 5x² + 2x − 7.

Step 1: Apply the power rule to each term. The power rule states that the derivative of xⁿ is nxⁿ⁻¹.

\frac{d}{dx}(x^n) = n\,x^{n-1}

Step 2: Differentiate the first term: 3x⁴.

\frac{d}{dx}(3x^4) = 3 \cdot 4x^3 = 12x^3

Step 3: Differentiate the second term: −5x².

\frac{d}{dx}(-5x^2) = -5 \cdot 2x = -10x

Step 4: Differentiate the third term: 2x. The derivative of 2x is simply 2.

\frac{d}{dx}(2x) = 2

Step 5: Differentiate the constant term: −7. The derivative of any constant is 0.

\frac{d}{dx}(-7) = 0

Step 6: Combine all the results.

f'(x) = 12x^3 - 10x + 2

Answer: The first derivative is f'(x) = 12x³ − 10x + 2.

Another Example

Problem: Find the first derivative of g(x) = sin(x) + 4x³ and evaluate it at x = 0.

Step 1: Differentiate sin(x). The derivative of sin(x) is cos(x).

\frac{d}{dx}(\sin x) = \cos x

Step 2: Differentiate 4x³ using the power rule.

\frac{d}{dx}(4x^3) = 12x^2

Step 3: Combine the results to get the first derivative.

g'(x) = \cos x + 12x^2

Step 4: Evaluate at x = 0. Since cos(0) = 1 and 12(0)² = 0:

g'(0) = \cos(0) + 12(0)^2 = 1 + 0 = 1

Answer: The first derivative is g'(x) = cos(x) + 12x², and g'(0) = 1. This means the slope of g(x) at x = 0 is 1.

Frequently Asked Questions

What is the difference between the first derivative and the second derivative?

The first derivative f'(x) measures how the original function f(x) changes — it gives the slope or rate of change. The second derivative f''(x) measures how the first derivative itself changes, which tells you about the concavity (curving direction) of the original function. You get the second derivative by differentiating the first derivative.

What does the first derivative tell you about a graph?

The first derivative at a point gives the slope of the tangent line to the graph at that point. Where f'(x) > 0, the function is increasing. Where f'(x) < 0, the function is decreasing. Where f'(x) = 0, the function has a horizontal tangent, which may indicate a local maximum, local minimum, or inflection point.

First Derivative vs. Second Derivative

The first derivative f'(x) tells you the rate of change (slope) of the original function. The second derivative f''(x) tells you the rate of change of the slope — that is, whether the graph is concave up or concave down. If f'(x) describes velocity, then f''(x) describes acceleration.

Why It Matters

The first derivative is the central tool of differential calculus. It lets you find where functions increase or decrease, locate maximum and minimum values, and determine the slope at any point on a curve. In science and engineering, first derivatives model real-world rates like velocity, growth rates, and marginal cost.

Common Mistakes

Mistake: Forgetting that the derivative of a constant term is zero, leading to leftover constants in the first derivative.

Correction: Any standalone constant (like −7 or +3) vanishes when you differentiate. Only terms containing the variable survive.

Mistake: Confusing f'(x) = 0 with meaning the function equals zero.

Correction: When f'(x) = 0, the slope is zero — the function has a horizontal tangent at that point. The function's value f(x) can be anything; it is not necessarily zero.

Related Terms

Derivative — General concept; first derivative is the basic case
Second Derivative — The derivative of the first derivative
nth Derivative — Generalization to any order of differentiation
Power Rule — Most common rule for computing derivatives
Differentiation — The process of finding a derivative
Slope — First derivative gives the slope at a point
Critical Point — Point where the first derivative is zero or undefined