Differentiating standard functions
Differentiation
- A-level
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
-
Differentiate $4x^3 - 3x^2 + 7x - 2$ with respect to $x$.
3 marks
-
Differentiate $e^{4x}$ and $\cos 3x$ with respect to $x$.
3 marks
-
Find the gradient of the curve $y = x^3 - 2x$ at the point where $x = 2$.
4 marks
-
Find the equation of the tangent to $y = x^2 - 3x$ at the point where $x = 4$.
4 marks
-
Differentiate $3e^{2x} + 4\sin x - \cos 2x$.
5 marks
-
The curve $y = x^3 + x$ has a tangent that is parallel to the line $y = 4x - 1$. Find the coordinates of the points of contact.
5 marks
-
Find the equation of the normal to the curve $y = e^{2x}$ at the point where $x = 0$.
6 marks
-
The tangent to the curve $y = x^3 - 6x$ has gradient $6$. Find the equations of the possible tangents.
7 marks
Mark scheme
-
Answer. $12x^2 - 6x + 7$
- Differentiate term by term.
-
Answer. $4e^{4x}$ and $-3\sin 3x$
- The derivative of $e^{4x}$ is $4e^{4x}$.
- The derivative of $\cos 3x$ is $-3\sin 3x$.
-
Answer. $10$
- $\dfrac{dy}{dx} = 3x^2 - 2$.
- At $x = 2$ the gradient is $12 - 2 = 10$.
-
Answer. $y = 5x - 16$
- When $x = 4$, $y = 4$, and the gradient is $5$.
- $y - 4 = 5(x - 4)$.
-
Answer. $6e^{2x} + 4\cos x + 2\sin 2x$
- The derivative of $3e^{2x}$ is $6e^{2x}$.
- The derivative of $4\sin x$ is $4\cos x$.
- The derivative of $-\cos 2x$ is $2\sin 2x$.
-
Answer. $(1, 2)$ and $(-1, -2)$
- $\dfrac{dy}{dx} = 3x^2 + 1 = 4$.
- $x = \pm 1$.
- The points are $(1, 2)$ and $(-1, -2)$.
-
Answer. $y = -\dfrac{1}{2}x + 1$
- The point is $(0, 1)$ and the gradient of the curve is $2$.
- The normal has gradient $-\dfrac{1}{2}$.
- $y - 1 = -\dfrac{1}{2}x$.
-
Answer. $y = 6x - 16$ and $y = 6x + 16$
- $3x^2 - 6 = 6$, so $x = \pm 2$.
- At $x = 2$ the point is $(2, -4)$ and the tangent is $y = 6x - 16$.
- At $x = -2$ the point is $(-2, 4)$ and the tangent is $y = 6x + 16$.