From tangent gradient to derivative

A tangent follows the direction of a smooth curve at a point. Its gradient gives the local rate of change. Differentiation produces a formula for that gradient at any permitted x-value; it does not give the height of the curve.

For \(y=ax^n\), where n is a positive integer, \(\dfrac{dy}{dx}=anx^{n-1}\). The derivative of a constant is zero. Differentiate each term of a sum separately; this course uses sums of at most three such terms with nonnegative integer powers.

Example

For \(y=2x^3-5x+4\), \(\dfrac{dy}{dx}=6x^2-5\). At \(x=2\), the gradient is 19, while the point’s y-coordinate is 10. These answer different questions.

An estimated tangent gradient can check the sign and approximate size of a derivative. Decreasing parts have negative gradient; horizontal tangents have zero gradient.