Differentiation

Topic overview

High-frequency lost marks in differentiation come from not first determining the form of the function. For implicit functions, parametric equations and tangent angles, you must first find the correct slope expression, then substitute the point.

Key concepts

When differentiating both sides of an implicit function with respect to $x$ , every term containing $y$ carries $dy/dx$ . When the slopes of the two lines are $m_1,m_2$ , the angle between them satisfies $\tan\theta=\left|\frac{m_2-m_1}{1+m_1m_2}\right|.$

Key points

◆Differentiate first, then substitute the coordinates
◆The $y$ in the product requires the product rule
◆Take the acute angle and check the perpendicular case

Worked examples

Try each example yourself before checking the explanation.

1Slope of an implicit function

If $x^2+xy+y^2=7$ , find the slope at the point $(1,2)$ .

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