Differentiation and integration part 1 Differentiating trig functions Gradient for parametric equations Gradient for a curve given implicitly Forming Differential equations Integrating

Differentiating trig functions You need to know

1. Differentiate Write

2.Differentiate Set out equations on the right of the page Differentiate each – take care with the names of the derivatives Combine Simplify

Example 3 A curve is given by (a)Find in terms of θ. Find the equation of the tangent to the curve at the point where the parameter takes the value

Example 3 The curve is given by

Implicit differentiation We use a version of the chain rule to help us to differentiate when the equation cannot easily be rewritten as y = …. This means we differentiate with the wrong letter and correct it by multiplying by dy/dx

Implicit Example 4 Differentiate the whole equation wrt x Rearrange

5.Implicit differentiation Find the stationary points on the curve given by Differentiate whole equation wrt x Collect terms and simplify Product rule needed for this term

Forming differential equations Newton’s law of cooling states that the rate of cooling of a body is proportional to the excess temperature. Using write this as a differential equation. The minus indicates decrease in temperature The dt derivative is always the rate of …. Excess temperature The k converts “proportional to” into an equation

Which of the functions on the table are of this type? Integrals of the form Once you recognise the fraction as being of this type, you can just write down the answer. Which of the functions on the table are of this type?