Differentiation Flashcards
What does differentiation represent?
The rate of change or the gradient of a curve at a point.
What is the derivative of f(x) = xⁿ?
f’(x) = nxⁿ⁻¹
What does dy/dx mean?
It is the derivative of y with respect to x.
What is the derivative of a constant?
What is the derivative of f(x) = sin(x)?
cos(x)
What is the derivative of f(x) = cos(x)?
-sin(x)
What is the derivative of f(x) = eˣ?
eˣ
What is the derivative of f(x) = ln(x)?
1/x
What is the product rule?
d(uv)/dx = u(dv/dx) + v(du/dx)
What is the quotient rule?
d(u/v)/dx = (v(du/dx) - u(dv/dx)) / v²
What is the chain rule?
dy/dx = dy/du × du/dx
What is a stationary point?
Where the derivative is 0 — i.e. the gradient is horizontal.
How do you classify a stationary point?
Use the second derivative: if >0 it’s a minimum, <0 it’s a maximum, =0 could be a point of inflection.
What is the second derivative?
The derivative of the derivative; it tells you about the curve’s concavity.
What is the derivative of y = axⁿ + bx + c?
dy/dx = anxⁿ⁻¹ + b
What does a positive second derivative indicate?
The graph is concave up (U-shaped).
What does a negative second derivative indicate?
The graph is concave down (n-shaped).
What does it mean if f’‘(x) = 0?
It may be a point of inflection — check sign change or use higher derivatives.
How do you find the gradient of a curve at a point?
Differentiate and substitute the x-value.
What is the purpose of differentiation in modelling?
To find maxima/minima or rates of change in real-world problems.
