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Properties and calculus

Differentiate sin⁡−1x\sin^{-1}x, cos⁡−1x\cos^{-1}x and tan⁡−1x\tan^{-1}x and apply the chain rule; evaluate integrals that yield inverse trigonometric forms including sin⁡−1 ⁣(x/a)\sin^{-1}\!(x/a) and (1/a)tan⁡−1 ⁣(x/a)(1/a)\tan^{-1}\!(x/a).

Worked examples

Chain rule differentiation

Straightforward

Problem

Differentiate y=sin⁡−1(3x2)y = \sin^{-1}(3x^2) and find dydx\dfrac{dy}{dx} at x=13x = \dfrac{1}{3}.

Definite integral using the standard form

Moderate

Problem

Evaluate ∫0319+x2 dx\displaystyle\int_0^3 \frac{1}{9 + x^2}\,dx.

Product rule with an inverse trig function

Challenging

Problem

Differentiate g(x)=x2sin⁡−1(x)g(x) = x^2 \sin^{-1}(x), then evaluate g′(0)g'(0) and g′ ⁣(12)g'\!\left(\dfrac{1}{2}\right).

Practise

Q1·Straightforward
The derivative of sin⁡−1(x)\sin^{-1}(x) is ddxsin⁡−1(x)=11−x2\dfrac{d}{dx}\sin^{-1}(x) = \dfrac{1}{\sqrt{1-x^2}}.

Evaluate this derivative at x=0x = 0.
Q2·Straightforward
The derivative of tan⁡−1(x)\tan^{-1}(x) is ddxtan⁡−1(x)=11+x2\dfrac{d}{dx}\tan^{-1}(x) = \dfrac{1}{1 + x^2}.

Evaluate this derivative at x=1x = 1.
Q3·Straightforward
Differentiate f(x)=tan⁡−1(2x)f(x) = \tan^{-1}(2x) using the chain rule.

Find f′(0)f'(0).
Q4·Moderate
Differentiate y=cos⁡−1(1−x2)y = \cos^{-1}(1 - x^2) and find dydx\dfrac{dy}{dx} at x=1x = 1.
Q5·Moderate
Evaluate ∫01/211−x2 dx\displaystyle\int_0^{1/2} \frac{1}{\sqrt{1 - x^2}}\,dx.

Give your answer in radians to 4 decimal places.
Q6·Moderate
Evaluate ∫0111+x2 dx\displaystyle\int_0^1 \frac{1}{1 + x^2}\,dx.

Give your answer in radians to 4 decimal places.
Q7·Moderate
Evaluate ∫0311+x2 dx\displaystyle\int_0^{\sqrt{3}} \frac{1}{1 + x^2}\,dx.

Give your answer in radians to 4 decimal places.
Q8·Challenging
Differentiate y=sin⁡−1 ⁣(x3)y = \sin^{-1}\!\left(\dfrac{x}{3}\right) and find the exact value of dydx\dfrac{dy}{dx} at x=0x = 0.

Express your answer as a fraction p/qp/q in lowest terms.
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Q9·Challenging
Evaluate ∫0114−x2 dx\displaystyle\int_0^1 \frac{1}{\sqrt{4 - x^2}}\,dx.

Give your answer in radians to 4 decimal places.

**Hint:** Use ∫1a2−x2 dx=sin⁡−1 ⁣(xa)+C\displaystyle\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{-1}\!\left(\frac{x}{a}\right) + C.
Q10·Challenging
Evaluate ∫0214+x2 dx\displaystyle\int_0^2 \frac{1}{4 + x^2}\,dx.

Give your answer in radians to 4 decimal places.

**Hint:** Use ∫1a2+x2 dx=1atan⁡−1 ⁣(xa)+C\displaystyle\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right) + C.
Q11·Challenging
Differentiate f(x)=xtan⁡−1(x)f(x) = x\tan^{-1}(x) and find f′(1)f'(1).

Give your answer to 4 decimal places.
Q12·Challenging
Differentiate y=cos⁡−1 ⁣(1x)y = \cos^{-1}\!\left(\dfrac{1}{x}\right) for x>1x > 1 and find dydx\dfrac{dy}{dx} at x=2x = 2.

Give your answer to 4 decimal places.