Derivatives of Inverse Trig Functions –Formulas, Rules & Examples

The derivatives of inverse trig functions are the derivatives of the inverse of basic trigonometric ratios such as sin, cos, and tan. The derivative of an inverse trig function represents the instantaneous rate of change of the inverse function for its input. The derivatives inverse functions are not trigonometric functions themselves, but rather algebraic functions involving square roots and powers of the input.

Let’s first understand the inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the inverses of the standard trigonometric ratios like sine, cosine, and tangent. They are used to find an angle when a trigonometric ratio is known. These inverse ratios are represented as;

\arcsin(x) \quad \text{or} \quad \sin^{-1}(x)

\arccos(x) \quad \text{or} \quad \cos^{-1}(x)

\arctan(x) \quad \text{or} \quad \tan^{-1}(x)

\text{arccsc}(x) \quad \text{or} \quad \csc^{-1}(x)

\text{arcsec}(x) \quad \text{or} \quad \sec^{-1}(x)

\text{arccot}(x) \quad \text{or} \quad \cot^{-1}(x)

What are Inverse Trig Derivatives

The inverse trig derivatives, also known as the arc derivatives, can be defined as the derivatives of the inverse of trigonometric functions. The implicit differentiation of trig functions is used to find the derivatives of inverse trigonometric ratios.

Representation of Derivatives of Inverse Trig Functions

There are six inverse trigonometric ratios. Differentiation is helpful in finding the derivatives of inverse trig functions. Derivatives show the rate of change of a function with the change of a variable. The derivatives of inverse trig functions are represented as;

Derivative \quad of \quad \sin^{-1}(x)

\frac{d}{dx} \left( \sin^{-1}(x) \right) = \frac{1}{\sqrt{1 - x^2}} \quad \text{for} \quad x \in (-1, 1)

Derivative \quad of \quad \cos^{-1}(x)

\frac{d}{dx} \left( \cos^{-1}(x) \right) = \frac{1}{\sqrt{1 - x^2}} \quad \text{for} \quad x \in (-1, 1)

Derivative \quad of \quad \tan^{-1}(x)

\frac{d}{dx} \left( \tan^{-1}(x) \right) = \frac{1}{{1 + x^2}} \quad \text{for} \quad x \in R

Derivative \quad of \quad \cosec^{-1}(x)

\frac{d}{dx} \left( \cosec^{-1}(x) \right) = -\frac{1}{|x| \sqrt{x^2 - 1}} \quad \text{for} \quad x \in\mathbb{R} \setminus [-1, 1]

Derivative \quad of \quad \sec^{-1}(x)

\frac{d}{dx} \left( \sec^{-1}(x) \right) =\frac{1}{|x| \sqrt{x^2 - 1}} \quad \text{for} \quad x \in\mathbb{R} \setminus [-1, 1]

Derivative \quad of \quad \cot^{-1}(x)

\frac{d}{dx} \left( \cot^{-1}(x) \right) = - \frac{1}{|x| \sqrt{x^2 - 1}} \quad \text{for} \quad x \in\mathbb{R}

Derivative of Inverse Trig Functions chart

Proof of Derivative of Inverse Trig Functions

We can find the derivative of inverse trigonometric functions using the implicit differentiation of trigonometric functions method. Let us look at the proof of derivative of inverse trig functions.

Derivative of Arcsin (sin⁻¹)

To find the derivative of Arcsin (sin-1), consider that

y = \sin^{-1}(x) \quad \Rightarrow \quad \sin(y) = x

Derivatives of Inverse Trig Functions

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What are Inverse Trig Derivatives

Representation of Derivatives of Inverse Trig Functions

Proof of Derivative of Inverse Trig Functions

Derivative of Arcsin (sin⁻¹)

FAQs

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Derivative of Arccos (Cos⁻¹)

Derivtive of Arctan (tan⁻¹)

Derivative of Arccsc (csc⁻¹)

Derivative of Arcsec (sec⁻¹)

derivative of Arccot (cot⁻¹)

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