9709 · 1.7
Differentiation flashcards
Revision flashcards for Cambridge 9709 Differentiation (syllabus 1.7). Flip, recall, then mark a real past-paper question.
Card
What is differentiation?
It is the process of finding the derivative, or gradient function, of a function. This new function tells you the instantaneous rate of change or the gradient of the original function at any point.
Card
What are the two main notations for the derivative of $y = f(x)$?
Leibniz's notation: $\frac{dy}{dx}$ (read as 'dee-why by dee-ex'). Lagrange's notation: $f'(x)$ (read as 'f-dashed-x' or 'f-prime-x').
Card
What is the Power Rule for differentiation?
For a function of the form $y = ax^n$, the derivative is $\frac{dy}{dx} = anx^{n-1}$. You multiply by the power and then reduce the power by one.
Card
What is the derivative of a constant term, e.g., $y = 7$?
The derivative of any constant is zero. A line $y=c$ is horizontal, so its gradient is always 0.
Card
How do you differentiate a function with multiple terms, like $y = x^3 + 4x^2$?
Differentiate each term separately and add or subtract them as in the original function. For $y = x^3 + 4x^2$, $\frac{dy}{dx} = 3x^2 + 8x$.
Card
What is a tangent to a curve?
A straight line that 'just touches' the curve at a single point. The gradient of the tangent is equal to the gradient of the curve at that point.
Card
What is a normal to a curve?
A straight line that is perpendicular to the tangent at the point of contact. Its gradient is the negative reciprocal of the tangent's gradient: $m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$.
Card
What is a stationary point (or turning point)?
A point on the curve where the gradient is zero. At this point, the tangent is horizontal. This occurs at local maxima, local minima, and points of inflection.
Card
How do you find the coordinates of a stationary point?
1. Differentiate the function to find $\frac{dy}{dx}$. 2. Set $\frac{dy}{dx} = 0$ and solve for $x$. 3. Substitute the $x$-value(s) back into the original equation for the curve to find the corresponding $y$-coordinate(s).
Card
How do you prepare an expression like $y = \frac{3}{x^2}$ for differentiation?
Rewrite it using negative indices before applying the power rule. $y = 3x^{-2}$. Then differentiate to get $\frac{dy}{dx} = 3(-2)x^{-2-1} = -6x^{-3} = -\frac{6}{x^3}$.
Card
How do you prepare an expression like $y = 5\sqrt{x}$ for differentiation?
Rewrite it using fractional indices before applying the power rule. $y = 5x^{1/2}$. Then differentiate to get $\frac{dy}{dx} = 5(\frac{1}{2})x^{1/2-1} = \frac{5}{2}x^{-1/2} = \frac{5}{2\sqrt{x}}$.