Differentiation Flashcards

All 13 cards in this deck

What does $\frac{dy}{dx}$ tell you about a curve?

It is the gradient function: it gives the gradient of the curve at any point, i.e. the rate of change of $y$ with respect to $x$.

The gradient of a curve at a particular point is equal to the gradient of what?

The gradient of the tangent to the curve at that point.

What are the steps to find the gradient of a curve at a given $x$-value?

e.g. find the gradient of $y = x^3 + 2x$ at $x = 2$

1. Differentiate: $\frac{dy}{dx} = 3x^2 + 2$
2. Substitute the $x$-value: $3 \times 2^2 + 2$
3. Gradient $= 14$

What are the steps to find the $x$-value at which a curve has a stated gradient?

e.g. find $x$ where $y = x^2 + 5x$ has gradient $11$

1. Differentiate: $\frac{dy}{dx} = 2x + 5$
2. Set the gradient function equal to the given value: $2x + 5 = 11$
3. Solve: $x = 3$

What does $\frac{d^2y}{dx^2}$ measure?

The rate of change of the gradient function — it is the derivative of $\frac{dy}{dx}$.

True or false? To find the $x$-values where a curve has zero gradient, you solve $\frac{dy}{dx} = 0$.

True. The gradient function is set to zero and solved as an equation.
e.g. $y = x^2 - 6x$ gives $2x - 6 = 0$.

What is the derivative of $kx^n$?

e.g. $y = 3x^5$

$\frac{dy}{dx} = knx^{n-1}$ — multiply by the power, then subtract 1 from the power.
e.g. $\frac{dy}{dx} = 15x^4$

What are the steps to differentiate a term with $x$ in the denominator?

e.g. $y = \frac{3}{x^2}$

1. Rewrite in index form: $y = 3x^{-2}$
2. Multiply by the power, subtract 1: $3 \times (-2)x^{-3}$
3. $\frac{dy}{dx} = -6x^{-3}$ (or $-\frac{6}{x^3}$)

What are the steps to differentiate a product of two brackets?

e.g. $y = (3x + 2)(x - 3)$

1. Expand the brackets: $y = 3x^2 - 7x - 6$
2. Differentiate each term: $\frac{dy}{dx} = 6x - 7$

What are the steps to differentiate an algebraic fraction with a single power of $x$ on the bottom?

e.g. $y = \frac{4x^3 + x^7}{x^4}$

1. Divide each term on top by the denominator: $y = 4x^{-1} + x^3$
2. Differentiate each term: $\frac{dy}{dx} = -4x^{-2} + 3x^2$

What are the steps to find $\frac{d^2y}{dx^2}$?

e.g. $y = x^4 + 3x^2$

1. Differentiate once: $\frac{dy}{dx} = 4x^3 + 6x$
2. Differentiate the result again: $\frac{d^2y}{dx^2} = 12x^2 + 6$

What are the steps to find an unknown constant from given gradient information?

e.g. $y = x^4 - 3kx^2$ has gradient $23$ at $x = 2$

1. Differentiate: $\frac{dy}{dx} = 4x^3 - 6kx$
2. Substitute $x = 2$ and equate to the gradient: $32 - 12k = 23$
3. Solve: $k = 0.75$

True or false? Differentiating $y = x^{-3}$ gives $\frac{dy}{dx} = -3x^{-2}$.

False. You subtract 1 from the power, so $\frac{dy}{dx} = -3x^{-4}$.