Differentiate Powers of x

## Differentiate Powers of x

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.

## Example 1

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

## Example 2

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.

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