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# Summary

## 6.8 Summary (EMCHM)

• The limit of a function exists and is equal to $$L$$ if the values of $$f(x)$$ get closer to $$L$$ from both sides as $$x$$ gets closer to $$a$$.

$\lim_{x\to a} f(x) = L$
• Average gradient or average rate of change:

$\text{Average gradient } = \frac{f\left(x+h\right)-f\left(x\right)}{h}$
• Gradient at a point or instantaneous rate of change:

$f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$
• Notation

${f}'\left(x\right)={y}'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}[f\left(x\right)]=Df\left(x\right)={D}_{x}y$
• Differentiating from first principles:

$f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$
• Rules for differentiation:

• General rule for differentiation:

$\frac{d}{dx}\left[{x}^{n}\right]=n{x}^{n-1}, \text{ where } n \in \mathbb{R} \text{ and } n \ne 0.$
• The derivative of a constant is equal to zero.

$\frac{d}{dx}\left[k\right]= 0$
• The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

$\frac{d}{dx}\left[k \cdot f\left(x\right) \right]=k \frac{d}{dx}\left[ f\left(x\right) \right]$
• The derivative of a sum is equal to the sum of the derivatives.

$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right] + \frac{d}{dx}\left[g\left(x\right)\right]$
• The derivative of a difference is equal to the difference of the derivatives.

$\frac{d}{dx}\left[f\left(x\right) - g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right] - \frac{d}{dx}\left[g\left(x\right)\right]$
• Second derivative:

$f''(x) = \frac{d}{dx}[f'(x)]$
• Sketching graphs:

The gradient of the curve and the tangent to the curve at stationary points is zero.

Finding the stationary points: let $$f'(x) = 0$$ and solve for $$x$$.

A stationary point can either be a local maximum, a local minimum or a point of inflection.

• Optimisation problems:

Use the given information to formulate an expression that contains only one variable.

Differentiate the expression, let the derivative equal zero and solve the equation.