**Differentiation** evaluates the rate of change of a function, which is the same as the gradient on a graph. A gradient on a graph is given as `frac(text(change in )y)(text(change in )x)`. A differentiation is shown as `frac(dy)(dx)` - often said as *differentiate dy by dx*.

The process is to take a term `ax^n`

>multiply the coefficient by the power<

>subtract 1 from the power<

Note that if there is a constant in the expression, then that constant is dropped.

If `y = ax^n`, then `frac(dy)(dx) = anx^(n-1)`.

If a function `f(x)` is differentiated, it is shown as `f`(x)`. A function `f(x) = ax^n` would differentiate to `f`(x) = anx^(n-1)`.

The power, `n`, can be positive, negative, a fraction or a decimal.

A curve has an equation of `y = 5x^3 - 7x^2 + 3x -5`. Find `frac(dy)(dx)`.

For each term, multiply the coefficient by the power, and subtract 1 from the power.

`5x^3` becomes `3 xx 5x^(3-1) = 15x^2`

`-7x^2` becomes `2 xx -7x^(2-1) = -14x`

`3x` becomes `1 xx 3x^(1-1) = 3`

The contant -5 is dropped

`frac(dy)(dx) = 15x^2 -14x + 3`

Answer: `15x^2 - 14x + 3`

A curve has the equation `y=-2x^3+4x-7`. A tangent is drawn to the curve when `x=-2`, and another tangent is drawn to the curve when `x=2`. Describe the relationship between these two tangents.

The gradient can be found by differentiating the equation.

`frac(dy)(dx) = 3 xx -2x^(3 - 1) + 1 xx 4x^(1-1) = -6x^2 + 4`

Substitute the values of `x` from the coordinates into the equation.

Both gradients are the same, therefore the lines are parallel.

Answer: They are parallel.

Reason: `frac(dy)(dx) = -6x^2 +4`.

When `x=-2, frac(dy)(dx) = -6(-2)^2 + 4 = -20`

When `x=2, frac(dy)(dx) = -6(2)^2 + 4 =-20`

Both gradients are the same.

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