Differentiating powers of xEdexcel International A Level Maths: Subtopic test

1. 1
   A curve has equation $y=3x^4-2x^2+7x-5$.

   (a)

   Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

   [1 mark]

   • A$12x^3-2x+7$
   • B$12x^4-4x+7$
   • C$12x^3-4x+7$
   • D$12x^3-4x+7x$

   (b)

   Find the gradient of the curve at the point where $x=1$.

   [1 mark]

   • A$15$
   • B$3$
   • C$8$
   • D$23$

   (c)

   Find the gradient of the curve at the point where it crosses the $y$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=f(x)$, where $f(x)=(2x+5)(x-1)$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$2$
   • B$4x-3$
   • C$2x^2+3x$
   • D$4x+3$

   (b)

   Find the value of $x$ at which the gradient of the curve is zero.

   [1 mark]

   • A$\frac34$
   • B$-\frac34$
   • C$-\frac43$
   • D$\frac43$

   (c)

   The curve crosses the positive $x$-axis at the point $A$. Find the gradient of the curve at $A$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=\dfrac{x^2+5x-3}{3x}$ for $x\neq0$.

   (a)

   Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

   [3 marks]

   (b)

   Find the coordinates of the points on the curve where the gradient is $\frac43$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=\dfrac{(x+2)^2}{x}+\dfrac{6}{\sqrt{x}}$ for $x>0$.

   (a)

   (i) Show that $f(x)=x+4+4x^{-1}+6x^{-\frac12}$.
   (ii) Find $f'(x)$.
   (iii) Find the exact value of $f'(4)$.

   [6 marks]

   (b)

   (i) Find $f''(x)$.
   (ii) Hence show that the gradient of the curve $y=f(x)$ is increasing for all $x>0$.
   (iii) Find the exact rate of change of the gradient when $x=1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions