The product rule tells you how to differentiate two functions multiplied together. The derivative of f(x)g(x) is the derivative of the first times the second, plus the first times the derivative of the second: (fg)’ = f’g + fg’. It is not simply f′ times g′.
Position, velocity and acceleration are linked by derivatives. If s(t) gives an object’s position at time t, then its velocity is v(t) = s'(t) and its acceleration is a(t) = v'(t) = s”(t). Velocity tells how fast the object moves and in which direction. Acceleration tells how fast the velocity is changing.
Higher-order derivatives are derivatives of derivatives. The second derivative, f″(x), is the derivative of f′(x). It tells you how fast the slope is changing. The third derivative is the derivative of f″(x), and so on. To find them, differentiate again, one step at a time.
How to find the equation of a tangent line with the derivative and point-slope form, with worked examples, horizontal tangents and practice problems.
The power rule says the derivative of xⁿ is nxⁿ⁻¹: bring the exponent down in front, then subtract 1 from it. So the derivative of x⁵ is 5x⁴. It works for any real exponent n. Together with the constant, constant multiple and sum rules, it differentiates any polynomial in one step.
A function is differentiable at x = a when its derivative f′(a) exists, which means the limit of the difference quotient is a finite number. Differentiability fails at corners, cusps, vertical tangents and discontinuities. Every differentiable function is continuous, but a continuous function can still fail to be differentiable, as |x| does at 0.
Sketching the graph of a derivative means drawing f′ from the graph of f by reading slopes. Where f has a horizontal tangent, f′ = 0. Where f rises, f′ is above the x-axis, and where f falls, f′ is below it. Steeper parts of f give values of f′ farther from 0.
The definition of the derivative is f′(x) = the limit as h → 0 of (f(x + h) − f(x))/h, when this limit exists. The derivative f′(x) is a new function that gives the slope of the tangent line, or the instantaneous rate of change, of f at every x.
The instantaneous rate of change of f at x = a is how fast f is changing at that exact moment. It is the limit of average rates of change over shorter and shorter intervals, the limit of (f(a + h) − f(a))/h as h → 0. On a graph, secant line slopes approach the slope of the tangent line at (a, f(a)).
The epsilon-delta definition of a limit makes “f(x) gets close to L” precise. The limit of f(x) as x approaches a is L when, for every ε > 0, there is a δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. In words, any tolerance on the output can be met by keeping x close enough to a.