Differentiate y=x^2sinx/cos2x also tanh^-1(tanh1/2x)?

For the first problem, we have:

.....( x ^ 2 ) sin x

y = ____________

..... cos 2x

Use the quotient rule for derivatives:

To take the derivative of the numerator: use the product rule

d / dx [(x ^ 2)sin x] = 2x sin x + (x ^ 2) cos x

The derivative of the denominator is -2sin 2x

The derivative of the overall is:

(2x sin x + (x ^ 2) cos x)(cos 2x) + 4(x ^ 2)sin^2 (x)cos x

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cos^2 (2x)

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The next problem is equal to 1 / (2x), (because the tanh ^ -1 and tanh cancel...)

So, this derivative is the derivative of 0.5 * (x ^ -1), and we can use the power rule to get

0.5 * -1 * (x ^ -2) or

- 1 / (2 (x ^ 2))

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Let's say the increase of w is represented by w1 = 1.02w,

The decrease of s is s1 = 0.97s

The decrease of d is d1 = 0.99d

To input into y1, we get

[1.02w * (0.97) ^ 3 * (s ^ 3)] / [0.99 ^ 4 * (d ^ 4)]

or, to simplify,

y1 = 1.02 * (0.97 ^ 3) / (0.99 ^ 4)

After simplifying, we get that the answer is approximately

y1 = 0.969y.

Therefore, y decreases by approximately 3.089%.