Differentiate 9tan3x with respect to x.a)9tan3x (3 log9 sec2x)b)9tan3x (3 log3 sec2x)c)9tan3x (3 log9 secx)d)-9tan3x (3 log9 sec2x)Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Differentiating 9tan(3x) with respect to x requires the application of the chain rule. Let's break down the steps to find the correct answer.

Step 1: Apply the chain rule to the function 9tan(3x).
The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x). In this case, f(g(x)) = 9tan(3x), so we need to find the derivatives of both f and g.

Step 2: Find the derivative of f(g(x)).
Since f(g(x)) = 9tan(3x), the derivative of f(g(x)) with respect to g(x) is 9.

Step 3: Find the derivative of g(x).
The function g(x) = 3x, so the derivative of g(x) with respect to x is 3.

Step 4: Apply the chain rule.
Using the chain rule, the derivative of 9tan(3x) with respect to x is 9 * 3 = 27.

Therefore, the correct answer is option 'A', 9tan(3x) (3 log9(sec^2x)).

Explanation:
- The chain rule is applied to differentiate the given function.
- The derivative of the outer function f(g(x)) = 9tan(3x) is found, which is 9.
- The derivative of the inner function g(x) = 3x is found, which is 3.
- The chain rule is applied to get the final derivative as 9 * 3 = 27.
- The answer is expressed in the given format, 9tan(3x) (3 log9(sec^2x)).

Note: The other options (B, C, and D) are not correct because they either don't apply the chain rule correctly or have incorrect derivatives.

Answer

Consider y=9^tan⁡3x
Applying log on both sides, we get
log⁡y=log⁡9^tan⁡3x
Differentiating both sides with respect to x, we get

(∵ Using u.v = u′v + uv′)
(dy/dx) = y(3 sec²⁡⁡x.log⁡9+0)
(dy/dx) = 9^tan⁡3x (3 log⁡9 sec²⁡x)

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