First principles: polynomial with multiple terms

Questions asking students to use differentiation from first principles to prove the derivative of a polynomial with multiple variable terms (e.g., 2x³ + 3x, 2x² - 5x + 2, x³ - x, 3x² + 2x, x - x²).

10 questions · Moderate -0.1

1.07g Differentiation from first principles: for small positive integer powers of x

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OCR H240/01 2019 June Q6

6 marks Moderate -0.3

6 Let $\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x$. Use differentiation from first principles to show that $\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + 3$.

OCR MEI Paper 1 2020 November Q12

9 marks Standard +0.3

12 A function is defined by $\mathrm { f } ( x ) = x ^ { 3 } - x$.

By considering $\frac { f ( x + h ) - f ( x ) } { h }$, show from first principles that $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1$.
Sketch the gradient function $\mathrm { f } ^ { \prime } ( x )$.
Show that the curve $y = f ( x )$ has a single point of inflection which is not a stationary point.

Given that $y = 2x^2 - 5x$, find $\frac{dy}{dx}$ from first principles. [5]
Given that $y = \frac{16}{5}x^4 + \frac{48}{x}$, find the value of $\frac{dy}{dx}$ when $x = 16$. [3]

WJEC Unit 1 2023 June Q9

11 marks Moderate -0.3

Given that $y = x^2 - 3x$, find $\frac{dy}{dx}$ from first principles. [5]
The function $f$ is defined by $f(x) = 4x^{\frac{3}{2}} + \frac{6}{\sqrt{x}}$ for $x > 0$.
1. Find $f'(x)$. [2]
2. When $x > k$, $f(x)$ is an increasing function. Determine the least possible value of $k$. Give your answer correct to two decimal places. [4]