11.02 Differentiating x^n for integer n
Consider the function f \left( x \right) = x^{n}, where n is any positive integer.
Complete the following table:
| f(x) | f'(x) |
|---|---|
| x | |
| x^2 | |
| x^3 | |
| x^4 | |
| x^n |
Find the derivative of y = - 6 with respect to x.
Find the derivative of the following:
y = x^{ - 7 }
y = x - 9
y = 7 x^{ - 3 }
y = 5 x^{5}
y = x^{2} - x + 8
y = x^{4} + x^{5}
y = - \dfrac{1}{5} x^{3}
y = \dfrac{1}{2} x^{5} + \dfrac{1}{5} x^{8}
Consider the function y = \dfrac{1}{x^{2}}.
Rewrite the function in negative index form.
Hence, find the derivative of y = \dfrac{1}{x^{2}}.
Consider the function y = \dfrac{7}{x}.
Rewrite the function in negative index form.
Find the derivative, giving your answer with a positive index.
Differentiate each of the following, giving your answer with a positive index:
y = x^{ - 5 } + 7
y = x^{ - 5 } + x^{7}
y = \dfrac{1}{4 x^{3}}
y = \dfrac{x^{9} + 1}{x^{5}}
Differentiate the following functions:
y = \dfrac{- 2}{x^{5}}
f \left( x \right) = 2 x + 9
y = 5 x^{4} + 3 x^{2} + 4 x
y = 2 x^{3} - 3 x^{2} - 4 x + 13
f \left( x \right) = \dfrac{x^{8}}{8} + \dfrac{x^{5}}{5} - 3 x
f\left( x \right) = 3 x^{ - 6 } + \dfrac{x^{ - 3 }}{7}
y = \dfrac{24}{x^{5}} - \dfrac{30}{x^{4}}
y = \dfrac{- 5 x + 1}{8}
y = \dfrac{5 x^{2} - 4 x + 1}{3}
f \left( x \right) = \dfrac{2 x}{9} + 7
Differentiate y = 7 a x^{7} - 2 b x^{3}, where a and b are constants.
Differentiate y = \dfrac{2}{x^{a}} - \dfrac{3}{x^{b}}, where a and b are constants, giving your answer with a positive index.
Find the derivative of each of the following functions:
f \left( r \right) = \dfrac{2}{r} + \dfrac{r}{3}.
For each of the following:
Express the function in expanded form.
Find the derivative of the function.
y = \left( 6 x + 5\right) \left(x + 3\right)
y = 2 x^{2} \left( 7 x + 2\right)
y = \left(x + 4\right)^{2}
y = \dfrac{4}{9} \left( - 4 x - 8\right)
Differentiate the function f \left( x \right) = \left(x + 2\right)^{3} by expanding it first.
Find f' \left( 2 \right) if f' \left( x \right) = 4 x^{3} - 3 x^{2} + 4 x - 6.
Find f' \left( 4 \right) if f' \left( x \right) = \dfrac{\left(x^{2} - 6\right) \times \left(2\right) + \left( 2 x\right) \left( 2 x\right)}{\left(x^{2} - 6\right)^{2}}.
By considering the graph of f \left( x \right) = 2 x, find f'\left( - 5 \right).
By considering the graph of f \left( x \right) = 2 x - 3, find f'\left( - 4 \right).
Find the gradients of the following functions at x=2:
Find the gradient of f \left( x \right) = x^{5} - 3 x^{4} at the point \left(3, 0\right).
Consider the function f \left( x \right) = x^{2}.
How many points on the graph of f \left( x \right) = x^{2} have a gradient of 2?
Find the x-coordinate(s) of the point(s) at whichf \left( x \right) = x^{2} has a gradient of 2.
Consider the function f \left( x \right) = 6 x^{2} + 5 x + 2.
Find f' \left( x \right).
Find f' \left( 2 \right).
Find the x-coordinate of the point at which f' \left( x \right) = 41.
Find the x-coordinate of the point at which f \left( x \right) = 5 x^{2} has a gradient of 10.
Consider the function y = 2 x^{2} - 8 x + 5.
Find \dfrac{dy}{dx}.
Hence, solve for the value of x at which the gradient is 0.
Consider the function f \left( x \right) = x^{3} - 4 x.
Find f' \left( x \right).
Find f' \left( 4 \right).
Find f' \left( - 4 \right).
Find the x-coordinates of the points at which f' \left( x \right) = 71.
Find the x-coordinate(s) of the point(s) at which f \left( x \right) = x^{4} has a gradient of 108.
Find the x-coordinate(s) of the point(s) at which f \left( x \right) = - 3 x^{3} has a gradient of - 81.