We now want to combine what we know about instantaneous rates of change, to be able to calculate the instantaneous rate of change at a point using the derivative - a function giving us the gradient of the tangent at any point on the graph. Differentiating a function helps us find the gradient function, which can then be used to find the gradient of the tangent at any point along the function (which is also the instantaneous rate of change at that point).
Functions of the type $f(x)=x^n$f(x)=xn are called power functions. Below are some rules to differentiate these types of functions:
This suggests that the derivative of a polynomial of degree $n$n is of degree $n-1$n−1, for integer values of $n\ge1$n≥1.
Looking further at just examples of functions of the form $f(x)=x^n$f(x)=xn, we have:
$f(x)$f(x) | $f'(x)$f′(x) |
---|---|
$x$x | $1$1 |
$x^2$x2 | $2x$2x |
$x^3$x3 | $3x^2$3x2 |
$x^4$x4 | $4x^3$4x3 |
$x^5$x5 | $5x^4$5x4 |
Noticing the pattern in the table, we could make the conjecture that the derivative of $f(x)=x^n$f(x)=xn is $f'(x)=nx^{n-1}$f′(x)=nxn−1. This is in fact the case and we call this formula the power rule. The power rule applies to not only positive integer values of $n$n but for $n$n being any real number. Once we establish the rule we can use the power rule as a quick way to find the derivative.
For a function $f(x)=x^n$f(x)=xn, the derivative is $f'(x)=nx^{n-1}$f′(x)=nxn−1, for $n$n any real number.
Find the derivatives of the following functions using the power rule.
Think: The power rule tells us "bring the power to the front and then subtract one from the power".
(a) $f(x)=x^2$f(x)=x2
Thus, $f'(x)=2x$f′(x)=2x.
(b) $g(m)=m^4$g(m)=m4
Thus, $g'(m)=4m^3$g′(m)=4m3.
Find the gradient of $f\left(x\right)=x^4$f(x)=x4 at $x=2$x=2.
Denote this gradient by $f'\left(2\right)$f′(2).
Use the applet below to explore how the gradient of the tangent changes at different points along $y=x^2$y=x2. Then answer the questions that follow.
Which feature of the gradient function tells us whether $y=x^2$y=x2 is increasing or decreasing?
The gradient function is decreasing when $y=x^2$y=x2 is increasing, and increasing when $y=x^2$y=x2 is decreasing.
The gradient function is increasing when $y=x^2$y=x2 is increasing, and decreasing when $y=x^2$y=x2 is decreasing.
The gradient function is negative when $y=x^2$y=x2 is increasing, and positive when $y=x^2$y=x2 is decreasing.
The gradient function is positive when $y=x^2$y=x2 is increasing, and negative when $y=x^2$y=x2 is decreasing.
For $x>0$x>0, is the gradient of the tangent positive or negative?
Positive
Negative
For $x\ge0$x≥0, as the value of $x$x increases how does the gradient of the tangent line change?
The gradient of the tangent line increases at a constant rate.
The gradient of the tangent line increases at an increasing rate.
The gradient of the tangent line remains constant.
For $x<0$x<0, is the gradient of the tangent positive or negative?
Positive
Negative
For $x<0$x<0, as the value of $x$x increases how does the gradient of the tangent line change?
The gradient of the tangent line remains constant.
The gradient of the tangent line increases at a constant rate.
The gradient of the tangent line increases at an increasing rate.
Complete the following statement:
"For $y=x^2$y=x2, the gradient of the tangent line changes at a constant rate. This means the derivative $y'$y′ is a function."
cubic
linear
constant
quadratic