11.03 Differentiating x^n for non-integer n
Consider the function y = \sqrt{x}.
Rewrite the function in index form.
Hence, find the derivative of y = \sqrt{x}. Give your answer in surd form.
Differentiate y = t \sqrt{t} with respect to t.
Consider the function y = \dfrac{1}{\sqrt[4]{x}}.
Rewrite the function in negative index form.
Hence, find the derivative of y = \dfrac{1}{\sqrt[4]{x}}. Give your answer in positive index form.
Consider the function y = \dfrac{5 x \sqrt{x}}{4 x^{5}}.
Rewrite the function in simplified negative index form.
Hence, find \dfrac{dy}{dx}.
Differentiate the following functions, give your answers in positive index form:
y = x^{ - \frac{1}{5} }
y = x^{3} \sqrt{x}
y = x \sqrt{x^{5}}
y = x^{4\frac{1}{2}}
y = 100 x^{1\frac{1}{3}}
y = x - x^{3} + \sqrt[8]{x} + 3
y = x^{3} \sqrt{x} + 3 x^{5}
y = \dfrac{1}{\sqrt[4]{x}} - x^{7} + \pi
y = \dfrac{x + 1}{\sqrt[7]{x}}
y = \dfrac{14}{x \sqrt{x}}
y = x^{6} + \dfrac{1}{x^{\frac{1}{5}}}
y = x^{6} + x^{\frac{1}{5}}
Differentiate the following functions:
y = x^{0.6}
y = 3 x^{0.6}
y = 3x^{2.5}
y =1.2 x^{-0.8}
y = - 10 x^{ - 0.5 }
Differentiate the following functions, give your answers in negative index form:
y = 8 x^{\frac{7}{9}}
y = x^{\frac{1}{2}} + 8 x^{\frac{3}{4}}
y = x^{ - 3\frac{1}{5} }
y = 3 x^{ - \frac{6}{5} }
y = 9 x^{\frac{1}{3}}
y = \dfrac{1}{x \sqrt{x}}
Differentiate the following functions, give your answers in surd form:
y = \dfrac{14}{\sqrt{x}}
y = \dfrac{2}{\sqrt{x}}
y = 14 x^{2} \sqrt{x}
y = 18 x \sqrt{x}
y = 26 \sqrt{x}
y = 27 \sqrt[3]{x}
y = \sqrt[3]{x^{2}}
y = \sqrt{ 36 x}
y = \sqrt{ 81 x}
y = \dfrac{2}{x} \sqrt{x}
Consider the function f \left( x \right) = \left(\sqrt{x} + 10 x^{2}\right)^{2}.
Rewrite the function f \left( x \right) in expanded form, with all terms written as powers of x.
Hence, find the derivative f' \left( x \right).
Consider the function y = \dfrac{8 x^{2} + 6 x + 4}{\sqrt{x}}.
Rewrite the function so that each term is a power of x.
Hence, find the derivative of the function.
For each of the following:
Rewrite the function in expanded form.
Hence, find the derivative \dfrac{d y}{d x}.
y = \left(\dfrac{4}{x} + 2 \sqrt{x}\right) \left(5 + \dfrac{6}{\sqrt{x}}\right)
y = \left(\sqrt[5]{x} + 2 \sqrt{x}\right)^{2}
y = \left( 3 \sqrt{x} + \dfrac{2}{x}\right)^{2}
y = \left( 4 \sqrt{x} - \dfrac{1}{\sqrt{x}}\right) \left( 4 \sqrt{x^{3}} + \dfrac{1}{x}\right)
y = \left( 2 x + \dfrac{3}{x}\right) \left( 6 \sqrt{x} + 5\right)
Differentiate the following functions:
f \left( x \right) = \dfrac{3 x - 2 \sqrt{x}}{\sqrt{x}}
f \left( x \right) = \dfrac{4}{\sqrt{x}} - \dfrac{\sqrt{x^{3}}}{2}
f \left( x \right) = \left(x + 1\right) \left(\sqrt{x} + 1\right) \left(\dfrac{1}{x} + 7\right)
Consider the function y = \dfrac{\sqrt{x}}{x}.
Differentiate y = \dfrac{\sqrt{x}}{x}.
Hence, find the gradient of the tangent to the curve y = \dfrac{\sqrt{x}}{x} at the point where x = 16.
Find the gradient of f \left( x \right) = x^{3} - 2 x^{4} + \sqrt{x} at the point \left(4, - 446 \right).
The function f \left( x \right) = \dfrac{\sqrt{x}}{2} has a tangent at \left(4, 1\right). Find the gradient of this tangent.
Find the gradient of f \left( x \right) = \dfrac{6}{\sqrt{x}} at the point \left(25, \dfrac{6}{5}\right).
Find the x-coordinate of the point at which f \left( x \right) = \sqrt{x} has a gradient of 6.
If the gradient to the tangent to y = \sqrt{x} is \dfrac{1}{6} at the point A, find the coordinates of A.
Consider the function f \left( x \right) = 6 \sqrt{x}. If f' \left( a \right) = \dfrac{3}{2}, find a.