Exercise 29

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Differentiate with respect to [math]x[/math]:

Question 1(i) \[10^{x}\] Solution
Question 1(ii) \[2^{\sqrt{x}}\] Solution
Question 1(iii) \[3^{x+2}\] Solution
Question 1(iv) \[5^{\sqrt{\sin{x}}}\]
Question 1(v) \[5^{x} \cdot \sin{x}\]
Question 1(vi) \[\log_{2} {\left(\tan {x}\right)}\]
Question 1(vii) \[\log_{5} {\left(\ln {x}\right)}\]
Question 1(viii) \[\ln{\left(\log_{5} {x}\right)}\]
Question 1(ix) \[e^{\log_{2}{x}}\]
Question 1(x) \[x^{x}\]
Question 1(xi) \[\left(\sin{x}\right)^{\tan{x}}\]
Question 1(xii) \[x^{\ln{x}}\]
Question 1(xiii) \[x^{x\tan{x}}\]
Question 1(xiv) \[x^{\frac{1}{x}}\]


Find:

Question 2(i) \[\int \! {5^{2x}} \,\mathrm{d}x\] Solution
Question 2(ii) \[\int \! {x \cdot 3^{x^{2}}} \,\mathrm{d}x\] Solution
Question 2(iii) \[\int \! {\frac{1}{3^{2x}}} \,\mathrm{d}x\] Solution
Question 2(iv) \[\int \! {\frac{2^{x}}{3^{2x}}} \,\mathrm{d}x\]