Implicit differentiation Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Implicit differentiation questions. See exactly how to solve problems on implicit-differentiation, exponential.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find dydx\dfrac{dy}{dx} in terms of xx and yy for the curve x2+y2=25x^{2} + y^{2} = 25.

Worked solution

  1. Write down the relation

    x2+y2=25x^{2} + y^{2} = 25

    Start from the implicit equation.

  2. Differentiate both sides with respect to xx

    2dydxy+2x=02 \frac{dy}{dx} y + 2 x = 0

    Treat y as a function of x and use the chain rule.

  3. Rearrange to make dydx\frac{dy}{dx} the subject

    dydx=−xy\frac{dy}{dx} = - \frac{x}{y}

    Collect the dy/dx terms and divide.

Answer
−xy- \frac{x}{y}
Question 2
2 markseasy
Find dydx\dfrac{dy}{dx} in terms of xx and yy for the curve xy=12x y = 12.

Worked solution

  1. Write down the relation

    xy=12x y = 12

    Start from the implicit equation.

  2. Differentiate both sides with respect to xx

    dydxx+y=0\frac{dy}{dx} x + y = 0

    Treat y as a function of x and use the chain rule.

  3. Rearrange to make dydx\frac{dy}{dx} the subject

    dydx=−yx\frac{dy}{dx} = - \frac{y}{x}

    Collect the dy/dx terms and divide.

Answer
−yx- \frac{y}{x}
Question 3
2 markseasy
Find dydx\dfrac{dy}{dx} in terms of xx and yy for the curve x2−y2=7x^{2} - y^{2} = 7.

Worked solution

  1. Write down the relation

    x2−y2=7x^{2} - y^{2} = 7

    Start from the implicit equation.

  2. Differentiate both sides with respect to xx

    −2dydxy+2x=0- 2 \frac{dy}{dx} y + 2 x = 0

    Treat y as a function of x and use the chain rule.

  3. Rearrange to make dydx\frac{dy}{dx} the subject

    dydx=xy\frac{dy}{dx} = \frac{x}{y}

    Collect the dy/dx terms and divide.

Answer
xy\frac{x}{y}
Question 4
2 markseasy
Find dydx\dfrac{dy}{dx} in terms of xx and yy for the curve x2+4y2=8x^{2} + 4 y^{2} = 8.

Worked solution

  1. Write down the relation

    x2+4y2=8x^{2} + 4 y^{2} = 8

    Start from the implicit equation.

  2. Differentiate both sides with respect to xx

    8dydxy+2x=08 \frac{dy}{dx} y + 2 x = 0

    Treat y as a function of x and use the chain rule.

  3. Rearrange to make dydx\frac{dy}{dx} the subject

    dydx=−x4y\frac{dy}{dx} = - \frac{x}{4 y}

    Collect the dy/dx terms and divide.

Answer
−x4y- \frac{x}{4 y}
Question 5
2 markseasy
The curve x2+y2=25x^{2} + y^{2} = 25 passes through (3, 4)\left(3,\ 4\right). Find the gradient dydx\dfrac{dy}{dx} at this point.

Worked solution

  1. Differentiate both sides with respect to xx

    2dydxy+2x=02 \frac{dy}{dx} y + 2 x = 0

    Differentiate implicitly, treating y as a function of x.

  2. Make dydx\frac{dy}{dx} the subject

    dydx=−xy\frac{dy}{dx} = - \frac{x}{y}

    Collect the dy/dx terms and divide.

  3. Substitute the point to find the gradient

    dydx∣(3, 4)=−34\left.\frac{dy}{dx}\right|_{\left(3,\ 4\right)} = - \frac{3}{4}

    Put the coordinates into the gradient function.

Answer
−34- \frac{3}{4}

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