Inverse Hyperbolic Functions

Find $\frac{dy}{dx}$ if:

a. $y = \cosh^{-1}(5x - 7)$ b. $y = \operatorname{sech}^{-1}(\ln x)$ c. $y = \ln(\tanh^{-1} x)$ d. $y = \sinh^{-1}(x^{-3})$
Differentiate the following with respect to $x$:

a. $y = x^{2} \cosh^{-1}(6x^{2} - 7x^{-2})$ b. $y = \cos(\sinh^{-1} x^{6})$ c. $y = \frac{\sinh^{-1}(2x)}{\tanh^{2}(4x + x^{-2})}$

Show that: a. $$\int \frac{dx}{\sqrt{a^{2} + x^{2}}} = \ln(x + \sqrt{x^{2} + a^{2}}) + C$$ b. $$\int \frac{dx}{\sqrt{x^{2} - a^{2}}} = \ln(x + \sqrt{x^{2} - a^{2}}) + C$$ c. $$\int \frac{dx}{a^{2} - x^{2}} = \frac{1}{2a} \ln\left|\frac{a + x}{a - x}\right| + C$$
Evaluate the following integrals: a. $\int \sinh^{6} x \cosh(x) , dx$ b. $\int \cosh(2x - 3) , dx$ c. $\int \sqrt{\tanh x} \operatorname{sech}^{2} x , dx$ d. $\int \frac{dx}{\sqrt{1 + 9x^{2}}}$ e. $\int \frac{dx}{\sqrt{9x^{2} - 25}}$ f. $\int \frac{dx}{x\sqrt{1 + 4x^{2}}}$

Source: FAC1004 Tutorial 9 25-26.pdf