AP Calculus AB Implicit Differentiation Worksheet (PDF)

AP Calculus AB math worksheet: Implicit Differentiation – Maths PDF

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This implicit practice worksheet for AP Calculus AB covers implicit differentiation (Common Core AP CALC AB 3.2), with a free PDF to print.

These 21 problems practice implicit differentiation, slopes of curves, tangent lines, horizontal and vertical tangents and second derivatives (AP Calculus AB, topics 3.2). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use circles, ellipses and tilted curves, vertical tangents, second derivatives and situations such as a drone path and a running track. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 3.2

Implicit practice worksheet: exercises 1 to 10

Exercise 1 – Circle

The circle of radius 5.

Find \(\frac{dy}{dx}\) for \(x^{2}+y^{2}=25\) and then the slope at \((3,4)\) and at \((-3,4)\).

Exercise 2 – Product of x and y

Find \(\frac{dy}{dx}\) for \(xy=6\).

Exercise 3 – Sum of cubes

Find \(\frac{dy}{dx}\) for \(x^{3}+y^{3}=9\).

Exercise 4 – Mixed terms

Find \(\frac{dy}{dx}\) for \(x^{2}+xy+y^{2}=7\).

Exercise 5 – Sine of y

Find \(\frac{dy}{dx}\) for \(\sin y=x\).

Exercise 6 – Exponential in y

Find \(\frac{dy}{dx}\) for \(e^{y}=x+y\).

Exercise 7 – Semicubical parabola

Find \(\frac{dy}{dx}\) for \(y^{2}=x^{3}\).

Exercise 8 – Product and a cube

Find \(\frac{dy}{dx}\) for \(x^{2}y+y^{3}=10\).

Exercise 9 – Slope at a point

The tangent at (2, 1) has slope −5/4.

The point \((2,1)\) lies on \(x^{2}+xy+y^{2}=7\). Find the slope of the curve there.

Exercise 10 – Tangent line to an implicit curve

Find the tangent line to \(xy^{2}=12\) at \((3,2)\).

Exercises 11 and up

Exercise 11 – Horizontal and vertical tangents

The ellipse 9x² + 4y² = 36.

For the ellipse \(9x^{2}+4y^{2}=36\), find \(\frac{dy}{dx}\) and the points with horizontal and vertical tangents.

Exercise 12 – A shifted circle

The circle x² + y² − 4x + 6y = 12.

Find the points where \(x^{2}+y^{2}-4x+6y=12\) has a horizontal tangent.

Exercise 13 – Normal line

Find the normal line to \(x^{2}+y^{2}=25\) at \((3,4)\).

Exercise 14 – Logarithm in the equation

Find \(\frac{dy}{dx}\) for \(\ln(xy)=x\).

Exercise 15 – Sine of a product

Find \(\frac{dy}{dx}\) for \(\sin(xy)=x\).

Exercise 16 – Second derivative

Find \(\frac{d^{2}y}{dx^{2}}\) for \(x^{2}+y^{2}=25\).

Exercise 17 – Vertical tangents

The curve x² − xy + y² = 3.

Find the points on \(x^{2}-xy+y^{2}=3\) where the tangent is vertical.

Exercise 18 – Drone flight path

A drone flies along the circle \(x^{2}+y^{2}=100\) (distances in miles from the base). At the point \((6,8)\), find the slope of its path and the equation of the line it would follow if it flew straight.

Exercise 19 – Running track

A model of an elliptical running track is \(4x^{2}+9y^{2}=36\) (in hundreds of feet). Verify that \(\left(\frac{9}{5},\frac{8}{5}\right)\) is on the track and find the slope there.

Exercise 20 – Horizontal tangent of a tilted curve

Find the points where \(x^{2}+xy+y^{2}=7\) has a horizontal tangent.

Exercise 21 – Derivative of a fractional power

Differentiate \(y^{3}=x^{2}\) implicitly and show that the result agrees with the power rule for \(y=x^{2/3}\).

Key facts behind this implicit practice worksheet

Key facts

Differentiate every term with respect to x; a y term gets a factor dy/dx.
Use the product rule for terms such as xy, and the chain rule for sin y or e^y.
The derivative may contain both x and y; substitute a point on the curve to get a slope.
Horizontal tangents: numerator zero. Vertical tangents: denominator zero.

How to use this implicit practice worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the implicit practice worksheet as a PDF with the button below and keep it with your notes.

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