Find the derivative of the function given below by first taking the logarithms o

Question

Find the derivative of the function given below by first taking the logarithms of each side of the equation. Explain why the power rule for derivatives, also given below, cannot be used to find the derivative of the given function sin 8x y sin 8x du du power rule for derivatives nu dx Choose the correct derivative of y with respect to x for y J sin 8x) OA. 6 sin 8x) (8x cot 8x ln sin 8x 6x OB. 6 sin 8x) (8x cos 8x ln sin 8x 6x OC. 60 sin 8x) 6x cot 8x 8x ln (sin 8x OD 6x (8x cot 8x In cos 8x 6(sin 8x) Explain why the power rule for derivatives cannot be used to find the derivative of the given function. Select the correct choice below OA. The power rule for derivatives cannot be applied because the derivative of u with respect to x is not defined. OB. The power rule for derivatives cannot be applied because the rule assumes that the exponent of the function is a constant. OC. The power rule for derivatives cannot be applied because the variable u is not defined. OD. None of the above

Explanation / Answer

Solution:

Q1) Taking natural logs, we get

ln y= ln (sin8x)^(6x) =6x ln (sin8x)

Then take derivatives:

1/y* y' = 6[lnsin8x + x*1/sin8x*8cos8x]

           =6[lnsin8x + 8x cot8x]

so,

y'= y *6[ ln sin8x +8x cot8x] =6 (sin 8x)^(6x) [ 8xcot8x +ln sin8x]

So, the answer (A) is right.

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Q2) The answer is (B)

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