Implicit differentiation and a tangent

Differentiation

A-level Maths (9MA0)

ORIGINAL

Differentiate every term in an implicit curve, collect the derivative, then build the tangent.

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Question

The curve

x^2+xy+y^2=7

passes through

(1,2)

. Find the equation of the tangent to the curve at this point.

Every step worked, with the reasoning.

1
$2x+x\dfrac{dy}{dx}+y+2y\dfrac{dy}{dx}=0$
Differentiate implicitly; use the product rule on $xy$ and the chain rule on $y^2$ .
2
$(x+2y)\dfrac{dy}{dx}=-(2x+y)$
Collect the terms containing $\frac{dy}{dx}$ on one side.
3
$\dfrac{dy}{dx}=-\dfrac{2x+y}{x+2y}$
Divide by $x+2y$ to make the gradient explicit.
4
At $(1,2)$ , $\dfrac{dy}{dx}=-\dfrac{2(1)+2}{1+2(2)}=-\dfrac45$ .
Evaluate the gradient at the stated point.
5
$y-2=-\dfrac45(x-1)$ , so $4x+5y=14$
Use the point-gradient equation and rearrange.

Answer: $4x+5y=14$