IB Maths AA HL Ahl 5 15 Hl Further Derivatives and Integrals Questions

Practise IB Mathematics HL 5.15 by applying further derivatives and integrals methods to questions from the Question Bank.

Exam points

Differentiate tan x, sec x, cosec x, cot x, a^x, log_a x, arcsin x, arccos x and arctan x, applying the relevant derivative formula accurately.
Integrate these derivative forms, including linear composites and partial-fraction rearrangements of integrands, then verify by differentiation.

[Maximum number: 6]

A function f is defined by $f(x)=\arcsin \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$ .

Show that $f^{\prime}(x)=\frac{2 x}{\sqrt{x^{2}}\left(x^{2}+1\right)}$ for $x \in \mathbb{R}, x \neq 0$ .

attempting to use the quotient rule to find $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{x^{2}-1}{x^{2}+1}\right)$ $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{2}-1}{x^{2}+1}\right)=\frac{2 x\left(x^{2}+1\right)-2 x\left(x^{2}-1\right)}{\left(x^{2}+1\right)^{2}}\left(=\frac{4 x}{\left(x^{2}+1\right)^{2}}\right)$
attempting to use the chain rule to find $\frac{\mathrm{d}}{\mathrm{dx}}\left(\arcsin \left(\frac{x^{2}-1}{x^{2}+1}\right)\right)$
let $u=\frac{x^{2}-1}{x^{2}+1}$ and so $y=\arcsin u$ and $\frac{\mathrm{d} y}{\mathrm{~d} u}=\frac{1}{\sqrt{1-u^{2}}}$ $f^{\prime}(x)=\frac{1}{\sqrt{1-\left(\frac{x^{2}-1}{x^{2}+1}\right)^{2}}} \times \frac{4 x}{\left(x^{2}+1\right)^{2}}$ $=\frac{4 x}{\sqrt{\left(x^{2}+1\right)^{2}-\left(x^{2}-1\right)^{2}}} \times \frac{1}{\left(x^{2}+1\right)}$ $=\frac{4 x}{\sqrt{4 x^{2}}} \times \frac{1}{\left(x^{2}+1\right)}$ $=\frac{2 x}{\sqrt{x^{2}}\left(x^{2}+1\right)}$

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IB Maths AA HL Ahl 5 15 Hl Further Derivatives and Integrals Questions question 1