xercise6. Dierentiate the given function(a) f (x) = ln(cos x)(b) f (x) = cos (ln x)(c) f (x) = log3(x2 − 4)(d) g (x) = ln a − xa + x(e) g (x) = √x ln x(f) h(x) = ln(e−x + xe−x )7. If f (x) = xln x , nd f ′(e).8. Use Logarithmic Dierentiation to nd the derivative of the function(a) y = (3x − 7)4(8x2 − 1)3(b) y = xx(c) y = xsin x(d) y =√ x2 + 1x + 1

1) Simplify the given expressiona) e2 ln 5b) ln ex2c) ln ex2+1d) e2 ln x+cos(2) Solve the given equationa) eln x = 3b) ln e−3x = 3c) ln(2x + 1) = 3d) 2ex+2 = 5e) ln x + ln(x − 1) = ln 2f) 2e−0.2x − 2 = 83) Show that the given functions are inverses of each othera) f (x) = e2x and g (x) = ln √xb) f (x) = ex/2 and g (x) = 2 ln x

17.Find dydxfor the following functions. Do not simplify your answer.a) y = x4log4(x2 − 5)b) y =ln(sin(x))4xc) x = 7 + exyd) y = sec ( √x)e) y = etan(x3)f) y = (sin(x + 4))x2

Consider the following functions:𝑓(𝑥)=cos⁡(𝑥3−𝑥)ℎ(𝑥)=∣𝑥−3∣3𝑔(𝑥)=ln⁡(∣𝑥∣+3)𝑠(𝑥)=sin⁡3(𝑥)​ f(x)=cos(x 3 −x)h(x)=∣x−3∣ 3 g(x)=ln(∣x∣+3)s(x)=sin 3 (x)​ Which of the following is true?A. f is even, h and s are odd.B.h and s are odd, g is even.C. f and g are even, s is odd.D.g and f are even, h is odd.E.s is odd, f and h are even.

f(x) = ln(x7 + 4)f '(x)

l´ımx−→0ln(cos(3x))ln(cos(2x)) =ï 00ò= l´ımx−→0−3 sin(3x)cos(3x)−2 sin(2x)cos(2x)= l´ımx−→0−3 sin(3x) cos(2x)−2 sin(2x) cos(3x)=ï 00ò= 32 l´ımx−→03 cos(3x) cos(2x) − 2 sin(3x) sin(2x)2 cos(2x) cos(3x) − 3 sin(2x) sin(3x) = 32 · 32 = 94