A limit is a value that a function approaches as x approaches a particular value from either side.

What is a derivative?

A derivative (also known as a gradient function ) is a function that relates the gradient of another function to the value of x .

True or False? The derivative of a constant function is always zero .

True. The derivative of a constant function is always zero .

A tangent is a straight line that touches a curve at a point without crossing through it.

True or False? The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

True. The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

What is the derivative of a sum (or difference ) of functions equal to?

The derivative of a sum (or difference ) of functions is equal to the sum (or difference) of their individual derivatives.

What is a normal (or normal line ) with regard to a function?

A normal (or normal line ) is a straight line that passes through a point on a curve and is perpendicular to the tangent at that point.

What is the relationship between the gradients of a tangent and a normal at the same point on a curve?

At the same point on a curve, the product of the gradients of a tangent and its normal is equal to -1.

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Differentiation (DP IB Analysis & Approaches (AA))

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Cards in this collection (19)

What is a limit?
A limit is a value that a function approaches as x approaches a particular value from either side.
What is a derivative?
A derivative (also known as a gradient function) is a function that relates the gradient of another function to the value of x.
What does $\frac{d y}{d x}$ mean?
$\frac{d y}{d x}$ means the derivative of y with respect to x.
What does $f^{'} (x)$ mean?
$f^{'} (x)$ means the derivative of the function $f (x)$ with respect to $x$ .
True or False?
The derivative of a constant function is always zero.
True.
The derivative of a constant function is always zero.
State the formula for differentiating $x^{n}$ .
If $f (x) = x^{n}$ , then $f^{'} (x) = n x^{n - 1}$ .
I.e. the original power comes in front as a multiplier, then the original power is reduced by 1.
This formula is in the exam formula booklet.
What is the derivative of $a x$ , where $a$ is a constant?
The derivative of $a x$ , where $a$ is a constant, is $a$ .
What is a tangent?
A tangent is a straight line that touches a curve at a point without crossing through it.
True or False?
The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.
True.
The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.
What is an alternative notation for $f^{'} (x)$ ?
If $y = f (x)$ , then an alternative notation for $f^{'} (x)$ is $\frac{d y}{d x}$ .
True or False?
The derivative of $\frac{4}{x}$ is $4 x^{- 2}$ .
False.
The derivative of $\frac{4}{x}$ is $- 4 x^{- 2}$ , which is the same as $- \frac{4}{x^{2}}$ .
To differentiate a reciprocal, first rewrite it as a negative power: $\frac{4}{x} = 4 x^{- 1}$ .
Then use "If $f (x) = x^{n}$ , then $f^{'} (x) = n x^{n - 1}$ ".
What is the derivative of a sum (or difference) of functions equal to?
The derivative of a sum (or difference) of functions is equal to the sum (or difference) of their individual derivatives.
True or False?
The formula for differentiating powers of $x$ , that says if $f (x) = x^{n}$ then $f^{'} (x) = n x^{n - 1}$ , can only be used when the power $n$ is an integer.
False.
The formula for differentiating powers of $x$ , that says if $f (x) = x^{n}$ then $f^{'} (x) = n x^{n - 1}$ , can be used when the power $n$ is any rational number.
What does it mean if $f^{'} (x) > 0$ ?
If $f^{'} (x) > 0$ , it means the function $f (x)$ is increasing.
True or False?
A function is decreasing if $f^{'} (x) < 0$ .
True.
A function is decreasing if $f^{'} (x) < 0$ .
What is a normal (or normal line) with regard to a function?
A normal (or normal line) is a straight line that passes through a point on a curve and is perpendicular to the tangent at that point.
What is the relationship between the gradients of a tangent and a normal at the same point on a curve?
At the same point on a curve, the product of the gradients of a tangent and its normal is equal to -1.
True or False.
The equation of the tangent to $y = f (x)$ at the point $(x_{1}, y_{1})$ can be found by using $y - y_{1} = f (x_{1}) (x - x_{1})$ .
False.
To find the equation of the tangent you need to use the value of the derivative (gradient function) of the curve at the point $(x_{1}, y_{1})$ .
The equation of the tangent to $y = f (x)$ at the point $(x_{1}, y_{1})$ can be found by using $y - y_{1} = f^{'} (x_{1}) (x - x_{1})$ .
What is an equation for the normal to a curve $y = f (x)$ at the point $(x_{1}, y_{1})$ ?
An equation for the normal to a curve $y = f (x)$ at the point $(x_{1}, y_{1})$ is $y - y_{1} = - \frac{1}{f^{'} (x_{1})} (x - x_{1})$ .

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