Differentiation Flashcards
given a curve with equation y=f(x), an expression for f’(x) is called…
the derivative
how to differentiate
multiply by the power and reduce the power by 1
for an expression of the form y=…, we denote the derivative with respect to x by
dy/dx
it is important before you differentiate that…
all brackets are multiplied out and there are no fractions with an x term in the denominator
how else can you write the square root of x?
x to the power of a half
how else to write 1/x squared
x to the power of negative 2
in a fractional indice, what are the numerator and demoninator?
numerator - power
denominator - root
how to differentiate an expression with several terms
differentiate each term seperately
the derivative of an x term is always…
constant
eg: 6x = 6
the derivative of a constant (eg: 3, 20, π) is always
zero
what can x/5 also be written as?
1/5 x
how to differentiate a fraction with multiple terms on the numerator
split the fraction up
how to denote answer if you are differentiating with respect to t?
d/dt
when differentiating with respect to a certain variable, all other variables are…
treated as constants
what is the derivative of px squared with respect to p?
x squared
the derivative of a function describes its…
rate of change
how to find the rate of change of a function when x=3
differentiate the function and then sub 3 in for x
what is the velocity v of an object defined as?
the rate of change of displacement s with respect to time t
d=ds/dt
acceleration a is defined as the…
rate of change of velocity with respect to time
a=dv/dt
we can determine the gradient of a curve, at a particular point, by…
considering a straight line which touches the curve at a point
the tangent
to work out the equation of a tangent we need to know which two things?
a point, of which at least one coordinate is given
the gradient, which is calculated by differentiating and substituting in the value of x at the required point.
when finding the equation of the tangent to a curve, dy/dx=….
gradient
steps on how to find the equation of a tangent to a curve?
- differentiate the equation of the curve
- sub in the value of x given in the question to the gradient expression
- if necessary, use x value in the question to work out y value by subbing into equation
- y-b=m(x-a)
how to find coordinates of points on curve where the tangent has a certain gradient?
- differentiate to get gradient of tangent
- find where it is equal to specified value
- find y coords using equation and subbing in calculated values of x
when is a curve said to be strictly increasing?
when dy/dx > 0
because tangent will slope upwards from left to right as gradient is positive
when is a curve said to be strictly decreasing?
when dy/dx < 0
because tangent will slope downwards from right to left as gradient is negative
how to determine whether a curve is increasing or decreasing at a given value of x
- differentiate
- sub in x value
- evaluate if < or > than 0
how to prove that a curve is never decreasing
if the expression for the derivative is (something)^2 because the result of squaring a number is always greater than or equal to zero
dy/dx > or equal to 0 for all values of x
at some points, a curve may be neither increasing nor decreasing - we call these points…
stationary points
what are the four possible stationary points?
- maximum turning point
- minimum turning point
- rising point of inflection
- falling point of inflection
how to show where an expression for a straight line is increasing or decreasing
- differentiate
- identify y-intercept
- calculate where it cuts the x-axis (when y=0)
- draw sketch
steps to determine the nature of stationary points
- differentiate the function
- state that stationary points occur when f’(x) = 0 and solve for x coords of stationary points
- find y coords by subbing x values into original equation
- use nature table to calculate f’(x) for values slightly lower and higher than sps
- evaluate nature of stationary points based on if the values are positive or negative
in order to sketch a curve, what needs to be found first?
- roots (when y=0)
- y-intercept (when x=0)
- stationary points and their nature
d/dx (sin x) =
cos x
d/dx (cos x) =
-sin x
the rules for differentiating sin and cos only work if…
in radians
the chain rule
differentiate the outer functions, the bracket stays the same, then multiply by the derivative of the bracket
closed intervals: the maximum and minimum y-values can either be…
- stationary points
2. end points of the closed interval
how to work out points at closed intervals
sub x values given in the question into original equation
things to remember when drawing a derived graph
- all stationary points on the original curve become roots
- wherever the original curve is strictly decreasing, the derivative is negative and will lie below the x-axis
- wherever the original curve is strictly increasing, the derivative is positive and will lie above the x-axis
what does the derived graph of a quadratic look like?
a linear (straight line)
what does the derived graph of a cubic look like?
a quadratic
what does the derived graph of a quartic look like?
a cubic
what is the process of finding optimal values known as?
optimisation
steps for optimisation questions
- differentiate
- =0 for sps
- find x value(s) for sps
- nature table
- if you get two answers, decide which one would not be possible (ie: negative look at diagram)
