Differentiate after index conversion

A question is this type if and only if it requires first converting an expression to index form (typically Ax^p + Bx^q) and then differentiating with respect to x.

8 questions · Moderate -0.5

1.07i Differentiate x^n: for rational n and sums

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Edexcel C12 2016 January Q12

10 marks Moderate -0.8

12. $$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$

Show that $\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C$, where $A , B , C$ and $k$ are constants to be determined.
Hence find $\mathrm { f } ^ { \prime } ( x )$.
Find an equation of the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $x = 4$ 2. LIIIII

Edexcel C1 2009 June Q9

8 marks Moderate -0.8

9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$

Show that $\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B$, where $A$ and $B$ are constants to be found.
Find $\mathrm { f } ^ { \prime } ( x )$.
Evaluate $\mathrm { f } ^ { \prime } ( 9 )$.

Edexcel C1 Q8

5 marks Moderate -0.8

8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$. END

Edexcel C1 Q8

11 marks Moderate -0.3

Given that

$$y = 2 x ^ { \frac { 3 } { 2 } } - 1$$

find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$,
show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 y = k$$ where $k$ is an integer to be found,
find $$\int y ^ { 2 } \mathrm {~d} x$$

AQA C2 2008 June Q1

9 marks Moderate -0.8

Write $\sqrt { x ^ { 3 } }$ in the form $x ^ { k }$, where $k$ is a fraction.
(1 mark)
A curve, defined for $x \geqslant 0$, has equation $$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find the equation of the tangent to the curve at the point where $x = 4$, giving your answer in the form $y = m x + c$.

AQA C2 2010 June Q6

13 marks Moderate -0.8

6 A curve $C$ has the equation $$y = \frac { x ^ { 3 } + \sqrt { x } } { x } , \quad x > 0$$

Express $\frac { x ^ { 3 } + \sqrt { x } } { x }$ in the form $x ^ { p } + x ^ { q }$.
1. Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find an equation of the normal to the curve $C$ at the point on the curve where $x = 1$.
1. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
2. Hence deduce that the curve $C$ has no maximum points. \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-7_1463_1707_1244_153}

AQA C2 2013 June Q6

12 marks Moderate -0.3

6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$

Express $\frac { 12 + x ^ { 2 } \sqrt { x } } { x }$ in the form $12 x ^ { p } + x ^ { q }$.
1. Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find an equation of the normal to the curve at the point on the curve where $x = 4$.
3. The curve has a stationary point $P$. Show that the $x$-coordinate of $P$ can be written in the form $2 ^ { k }$, where $k$ is a rational number.

Edexcel C2 Q6

11 marks Standard +0.3

6. Given that $\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0$,

find, to 3 significant figures, the value of $x$ for which $\mathrm { f } ( x ) = 5$.
Show that $\mathrm { f } ( x )$ may be written in the form $A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C$, where $A , B$ and $C$ are constants to be found.
Hence evaluate $\int _ { 1 } ^ { 2 } f ( x ) d x$.