Derivative of trigonometric functions like tan(x) often are explained using a rule (product or quotient) and a sequence of algebraic simplifications. The rule is great but if you are looking for intuitive approach, read on.

For better visual satisfaction I created interactive diagrams. Drag cursor over or tap on the diagrams.

Triangles are **similar** when they are scaled, flipped or rotated versions of each other.

For example, two similar triangles above are scaled and flipped versions of each other.

Trigonometry function produce same results for similar triangles.

A unit circle is a circle with radius of one and circumference of 2×π.

Radians describe angle as well as circumference within the angle. For example, 2×π — full circle and full circumference, π/4 — stright angle and quarter of circumference.

Here is a unit circle with sine and cosine for some angle x.

Sine is opposite over hypotenuse and cosine is adjecent over hypotenuse. Proportions in diagram above are result of hypotenuse being 1.

There is a convenient way to present tangens and secant on a unit circle.

Tangens is opposite over adjecent and secant is hypotenuse over adjecent. Proportions in diagram above are result of the adjecent side getting scaled up to 1.

There is a convenient way to present cotangens and cosecant on a unit circle.

Cotangens is adjecent over opposite and cosecant is hypotenuse over opposite. Proportions in diagram above are result of the opposite side getting scaled up to 1.

Let's draw sine and cosine on a unit circle.

Let there be a small increase in x, call it dx. Adding few lines forms a new triangle.

Observe when angle dx is infinitely small then the chord dx is nearly perpendicular to both **hypotenuse** lines. Then we can reason that angles of the new triangle are the same as of the original. It is just rotated and scaled down version of it, where hypotenuse got decreased from single unit down to dx!

The ratio between sides of a triangle never changes hence sine and cosine values are the same for the new triangle.

Increase in sine is marked in red in the new triangle. It's length is cos(x)×dx which means the derrivative of sin(x) is cos(x).

Decrease in cosine marked in blue in the new triangle. It's length is sin(x)×dx which means the derrivative of cos(x) is −sin(x).

Let's draw tangens and secant on a unit circle.

Let there be a small increase in x, call it dx. We can add few lines and form a new triangle as follows.

Observe when angle dx is infinitely small then the chord dx is nearly perpendicular to both **secant** lines. Then we can reason that angles of the new triangle are the same as of the original. It is just rotated and scaled down version of it, where hypotenuse got decreased from single unit down to sec(x)×dx!

Let's scale up the new triangle while keeping it on secant lines untill it aligns itself with tangens line. The resulting triangle remains similar to the original one.

The resulting adjecent side of the new triangle decreases to sec(x)×dx, and the rest decreases proportionally.

The ratio between sides of a triangle never changes hence tangens and secant values are the same!

Increase in tangens is marked in green in the new triangle. It's length is sec(x)×sec(x)×dx which means the derrivative of tan(x) is sec(x)×sec(x).

Increase in secant is marked in blue in the new triangle. It's length is tan(x)×sec(x)×dx which means the derrivative of sec(x) is tan(x)×sec(x).

Let's draw cotangens and cosecant on a unit circle.

Let there be a small increase in x, call it dx. We can add few lines to form a new triangle.

Observe when angle dx is infinitely small then the chord dx is nearly perpendicular to both **cosecant** lines. Then we can reason that angles of the new triangle are the same as of the original. It is just rotated and scaled down version of it, where hypotenuse got decreased from single unit down to csc(x)×dx!

Let's scale up the new triangle while keeping it on cosecant lines untill top edge of the unit circle. The resulting triangle remains similar to the original one.

The resulting opposite side of the new triangle decreases to csc(x)×dx, and the rest decreases proportionally.

The ratio between sides of a triangle never changes therefore cotangens and cosecant values are the same!

Decrease in cotangens is marked in green in the new triangle. It's length is csc(x)×csc(x)×dx which means the derrivative of cot(x) is −csc(x)×csc(x).

Decrease in cosecant is marked in red in the new triangle. It's length is cot(x)×csc(x)×dx which means the derrivative of csc(x) is −cot(x)×csc(x).

Thank you for reading.