10.05 Power rule (x^n)
Consider the graph of y = x:
Find the gradient of the line at x = 4.
Find the gradient at any value of x.
What can be said about the gradient of a linear function?
Consider the graph of y = x^{2} and its gradient function y = 2x:
What can be said of the sign of the gradient function when y = x^{2} is increasing or decreasing?
For x > 0, is the gradient of the tangent positive or negative?
For x \geq 0, as the value of x increases how does the gradient of the tangent line change?
For x < 0, is the gradient of the tangent positive or negative?
For x < 0, as the value of x increases how does the gradient of the tangent line change?
For y = x^{2}, the gradient of the tangent line changes at a constant rate. What type of function is the derivative of \\ y = x^{2}?
Consider the graph of y = x^{3} and its gradient function y = 3x^2.
For x > 0, is the gradient of the tangent positive or negative?
For x \geq 0, as the value of x increases how does the gradient of the tangent line change?
For x < 0, is the gradient of the tangent positive or negative?
For x < 0, as the value of x increases how does the gradient of the tangent line change?
For y = x^{3}, the gradient of the tangent line first decreases at a decreasing rate, then increases at an increasing rate. What type of function is the derivative of y = x^{3}?
Consider the graph of y = x^{4} and its gradient function y = 4x^3.
For x > 0, is the gradient of the tangent positive or negative?
For x \geq 0, as the value of x increases how does the gradient of the tangent line change?
For x < 0, is the gradient of the tangent positive or negative?
For x < 0, as the value of x increases how does the gradient of the tangent line change?
For y = x^{4}, the gradient of the tangent line is increasing, first at a decreasing rate and then at an increasing rate. What type of function is the derivative of \\ y=x^4?
Consider the functions f \left( x \right) = x^{5} and g \left( x \right) = x^{4}.
Sketch the graph of f \left( x \right) and its derivative.
Sketch the graph of g \left( x \right) and its derivative.
If the degree of a function is even, will the degree of its derivative function be odd or even?
If the degree of a function is odd, will the degree of its derivative function be odd or even?
Consider the function f \left( x \right) = x^{2}.
Find the derivative of f \left( x \right) = x^{2} from first principles.
The derivative of y = x^{n} for other values of n can be found using first principles in a similar way. The results for n = 2, 3, 4 and 5 are summarised in the following table:
| y | x^2 | x^3 | x^4 | x^5 |
|---|---|---|---|---|
| y' | 2x | 3x^2 | 4x^3 | 5x^4 |
By observing the pattern in the table, find the derivative of the function g \left( x \right) = x^{8}.
Write the general form of the derivative of y = x^{n} for any value of n.
For each of the following functions, state the degree of the gradient function:
y = x^{2}
y = x^{3}
y = x^{4}
Hence, state the degree of the derivative of a polynomial function of degree n.
Find the derivative of the following functions with respect to x:
Consider the function y = \sqrt{x}.
Rewrite the function in index form.
Find the derivative of y = \sqrt{x}, in surd form.
Consider the function y = \dfrac{1}{x^{2}}.
Rewrite the function in negative index form.
Find the derivative of y = \dfrac{1}{x^{2}}.
Consider the function y = \dfrac{1}{\sqrt[4]{x}}.
Rewrite the function in negative index form.
Find the derivative of y = \dfrac{1}{\sqrt[4]{x}}, in positive index form.
Differentiate y = t \sqrt{t}.
Differentiate y = x^{ - \frac{1}{5} } with respect to x. Express your answer in positive index form.
Find the gradient of f \left( x \right) = x^{4} at x = 2. Denote this gradient by f' \left( 2 \right).
Consider the function f \left( x \right) = x^{9}.
Find the gradient of the tangent to the function at x = 1.
Find the equation of the tangent to the function at x = 1.
Consider the function f \left( x \right) = \dfrac{1}{\sqrt{x}}.
Find the gradient of the tangent to the function at x = 1.
Find the equation of the tangent to the function at x = 1.
David draws the graphs of x^{2}, x^{3}, x^{4} and x^{5} and draws the tangents to each one at the point where x = 1.
David then notices where each of the tangents cut the y-axis and records this in the table below:
| \text{Graph} | y\text{-intercept of the tangent line} | \text{Gradient of the tangent} |
|---|---|---|
| y=x^2 | (0, -1) | |
| y=x^3 | \left(0, -2\right) | |
| y=x^4 | \left(0, -3\right) | |
| y=x^5 | \left(0, -4\right) |
Complete the table by calculating the gradient of each of the tangents at \\ x = 1.
Following the pattern in the table, what would be the gradient of the tangent to the graph of y = x^{n} at the point where x = 1?
Could the equation of the derivative of y = x^{6} be y' = x^{5}? Explain your answer.
The graph of y = x^{5} is shown below labelled as A. Fiona then graphs the derivative of the function, labelling it as B. She then finds the derivative of graph B to get graph C, then differentiates again to get graph D and differentiates again to get graph E.
State whether the following statements are true about this sequence of derivatives:
The derivative of a function is always positive when the function is negative, and negative when the function is positive.
Each graph is a function of the form a x^{n}.
For any value of x, the value of the derivative will always be greater than the value of the function.
The degree of the derivative is always different to the degree of the function.
Lucy used first principles to find the derivatives of y = x^{\frac{1}{2}}, y = x^{\frac{3}{2}} and y = x^{\frac{5}{2}}. Her results are shown in the following table:
Find the derivative of y = x^{\frac{4}{3}}.
Find the derivative of y = x^{\frac{3}{4}}.
| y | x^{\frac{1}{2}} | x^{\frac{3}{2}} | x^{\frac{5}{2}} |
|---|---|---|---|
| \dfrac {dy}{dx} | \dfrac{1}{2}x^{-\frac{1}{2}} | \dfrac{3}{2}x^{\frac{3}{2}} | \dfrac{5}{2}x^{\frac{3}{2}} |