Find \(f\textquotesingle(x)\) when \(f(x) = 12{x^{\frac{2}{3}}}\)
\[f\textquotesingle(x)=\frac{4}{3}{x^{-\frac{1}{3}}}\]
\[f\textquotesingle(x) = 8{x^{ - \frac{1}{3}}}\]
\[f\textquotesingle(x) = \frac{2}{3}{x^{ - \frac{1}{3}}}\]
Differentiate \(y = \sqrt x (x + 1)\)
\[\frac{{dy}}{{dx}} = \frac{3}{2}{x^{\frac{1}{2}}} + \frac{1}{2}{x^{ - \frac{1}{2}}}\]
\[\frac{{dy}}{{dx}}={x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}\]
\[\frac{{dy}}{{dx}} = \frac{1}{2}{x^{\frac{1}{2}}} - \frac{1}{2}{x^{ - \frac{1}{2}}}\]
Differentiate \(y = 4x - 2\sin x\)
\[\frac{{dy}}{{dx}} = 4 - 2\cos x\]
\[\frac{{dy}}{{dx}} = 4x - 2\cos x\]
\[\frac{{dy}}{{dx}} = 4 + 2\cos x\]
Differentiate \(y = 5\cos x - 1\)
\[\frac{{dy}}{{dx}} = 5\sin x\]
\[\frac{{dy}}{{dx}} = - 5\sin x - x\]
\[\frac{{dy}}{{dx}} = - 5\sin x\]
Find \(\frac{{dy}}{{dx}}\) when \(y = 7\sin x + \cos x\).
\[\frac{{dy}}{{dx}} = - 7\cos x - \sin x\]
\[\frac{{dy}}{{dx}} = 7\cos x + \sin x\]
\[\frac{{dy}}{{dx}} = 7\cos x - \sin x\]
Find \(\frac{{dy}}{{dx}}\) when \(y = {({x^2} + 5)^{11}}\)
\[\frac{{dy}}{{dx}} = 11{({x^2} + 5)^{10}}\]
\[\frac{{dy}}{{dx}} = 22x{({x^2} + 5)^{10}}\]
\[\frac{{dy}}{{dx}} = 22{(2x)^{10}}\]
Differentiate \(y = {(1 - x)^{ - \frac{1}{2}}}\)
\[\frac{{dy}}{{dx}} = \frac{1}{2}{(1 - x)^{ - \frac{3}{2}}}\]
\[\frac{{dy}}{{dx}} = - \frac{1}{2}{(1 - x)^{ - \frac{3}{2}}}\]
\[\frac{{dy}}{{dx}} = - \frac{1}{2}{(1 - x)^{ - \frac{1}{2}}}\]
Differentiate \(y = \cos (1 + 2x)\)
\[\frac{{dy}}{{dx}} = 2\cos (1 + 2x)\]
\[\frac{{dy}}{{dx}} = - 2\sin (1 + 2x)\]
\[\frac{{dy}}{{dx}} = 2\sin (1 + 2x)\]
Differentiate \(y = {(\cos 2x)^3}\)
\[\frac{{dy}}{{dx}} = - 6\sin 2x\,{\cos ^2}2x\]
\[\frac{{dy}}{{dx}} = - 2\sin 2x\,{\cos ^2}2x\]
\[\frac{{dy}}{{dx}} = - 3\sin 2x\,{\cos ^2}2x\]
Find the equation of the tangent to the curve with equation \(y = 3{x^2} + 2x - 5\) at the point where x = - 2
\[10x-y+17=0\]
\[8x-y+13=0\]
\[10x+y+17=0\]
At what point(s) on the curve with equation \(y = 5{x^3}\) is the tangent parallel to the line with equation \(15x - 4y + 8 = 0\)?
\(\left( {-\frac{1}{2},-\frac{5}{8}} \right)\) and \(\left( {\frac{1}{2},\frac{5}{8}} \right)\)
\[\left( {\frac{1}{2},\frac{5}{4}} \right)\]
\[\left( { - \frac{1}{4},\frac{5}{{16}}} \right)and\left( {\frac{1}{4},\frac{5}{{16}}} \right)\]