Differentiation: gradient, rate of change, tangents and normals

Learning objectives

Describe the derivative informally as the limit of the gradient of a chord as the two points move together, and interpret it as the gradient of the tangent at a point

Interpret a derivative as a rate of change in an applied context, stating its units as the units of the dependent variable per unit of the independent variable

Differentiate f(x) = ax^n to get f′(x) = anx^(n − 1) for integer n, and differentiate any sum of such terms, rewriting reciprocals such as 1/x² as x⁻² first

Evaluate the gradient of a curve at a given point, and find the coordinates of the point or points at which the gradient takes a specified value

Find the equation of the tangent to a curve at a point, and of the normal, using the fact that the normal gradient is the negative reciprocal of the tangent gradient

Use technology to find a numerical derivative at a point and to draw a tangent, and check an algebraic answer against it