Derivatives 1

L2MAEL

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Part 1

Find the derivative ($\frac{dy}{dx}$) of the following functions:

a)

$$y=x^2$$

b)

$$y=2x^4$$

c)

$$y=5x^2+10x+6$$

d)

$$y=-\frac{4}{3}{x}^9+5x+1$$

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Part 2

Find the derivative ($\frac{dy}{dx}$) of the following functions:

a)

$$y=\frac{1}{x}$$

b)

$$y=(x+2{)}^2$$

c)

$$y=(x+1{)}^4$$

d)

$$y=-(x-1{)}^2$$

e) note: $p$ is a constant.

$$y=p{x}^3+\frac{1}{p}{x}^2+p\sqrt{x}+p+1$$

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Part 3

a) Find the derivative ($\frac{dy}{dx}$) of the function $x^2+2x+1$, and state what the derivative means.

b) $\boxed{\text{2025Q1a}}$ Find the gradient of the following function at the point where $x=-0.5$:

$$y=2x^3-3x^2+\frac{x}{2}+7$$

c) $\boxed{\text{2024Q1a}}$ A function is given by $f(x)=4x^3-6x^2+5x+2$. Use calculus to find the gradient of the graph of the function at the point where $x=2$.

d) $\boxed{\text{Equation from 2025Q3d}}$ Find the derivative $\frac{dy}{dx}$ of the following function where $k$ is a positive constant:

$$y=\frac{x^4}{4}+\frac{(k-3)x^3}{3}-\frac{3k{x}^2}{2}+k$$