Interpret the derivative as a gradient and rate of change, and differentiate powers of x and the standard exponential, logarithmic and trigonometric functions

What this dot point is asking

SEAB wants you to understand the derivative as the gradient of a curve and the instantaneous rate of change of one quantity with respect to another, and to differentiate powers of $x$ and the standard exponential, logarithmic and trigonometric functions. This is the entry point to the whole calculus strand.

The answer

What the derivative means

The derivative $\dfrac{dy}{dx}$ measures how fast $y$ changes as $x$ changes. Graphically it is the gradient of the tangent to the curve $y = f(x)$ at a point. As a rate, it is the change in $y$ per unit change in $x$ at that instant.

The power rule

For any constant power $n$:

Multiply by the power, then reduce the power by one. A constant multiple stays attached ($\frac{d}{dx}(kx^n) = knx^{n-1}$), and the derivative of a constant is zero. Differentiate a sum term by term.

Rewriting before differentiating

Roots and reciprocals must be rewritten as powers first: $\sqrt{x} = x^{1/2}$ and $\dfrac{1}{x} = x^{-1}$. Then the power rule applies directly, and you tidy negative or fractional indices back at the end.

The standard derivatives

Three results to memorise:

Note the minus sign when differentiating cosine. The exponential $e^x$ is the one function equal to its own derivative, which is why it appears so often in growth and decay.

Notation and the second derivative

The derivative is written $\dfrac{dy}{dx}$, $f'(x)$ or $y'$, all meaning the same thing. Differentiating a second time gives the second derivative $\dfrac{d^2y}{dx^2}$ or $f''(x)$, which measures how the gradient itself is changing and is used later to classify turning points.

Differentiating sums and multiples

Differentiation is linear: the derivative of a sum is the sum of the derivatives, and a constant multiple simply carries through. So a polynomial is differentiated term by term, each term handled by the power rule, with no need to expand brackets unless that makes a term a recognisable power.

Examples in context

Example 1. Speed from a distance graph. If distance $s$ is a function of time $t$, the derivative $\dfrac{ds}{dt}$ is the speed, the gradient of the distance-time graph, which is exactly how kinematics uses differentiation.

Example 2. Slope of a hillside. A cross-section of terrain modelled by $y = f(x)$ has gradient $\dfrac{dy}{dx}$ giving the steepness at each point, so engineers read off where a slope is steepest by differentiating.

Try this

Q1. Differentiate $y = x^5$. [1 mark]

• Cue. $5x^4$.

Q2. Differentiate $y = 4\sqrt{x}$. [2 marks]

• Cue. $4 \cdot \tfrac{1}{2}x^{-1/2} = \dfrac{2}{\sqrt{x}}$.

Q3. Find $\dfrac{dy}{dx}$ for $y = e^x - 3\cos x$. [2 marks]

• Cue. $e^x + 3\sin x$.