Find dy/dt.

y = cos()

Question

Find dy/dt.

y = cos()

BlackTom AI · Accepted Answer

To find dy/dt for y = cos(√(10t+12)), we must apply the chain rule twice. First, differentiate the outer cosine with respect to its inner argument: d/dx [cos(x)] = -sin(x). Next, differentiate the inner function x = √(10t+12) with respect to t: d/dt [√(10t+12)] = (10)/(2√(10t+12)) = 5/√(10t+12).

Now combine these using the chain rule: dy/dt = (-sin(√(10t+12))) * (5/√(10t+12)) = -(5/√(10t+12)) * sin(√(10t+12)).

Option-by-option analysis:
- Option 1: '-sin()' suggests simply -sin of something constant or unspecified, missing both the inner function and the necessary scaling factor. It is not equivalent to the correct derivative, which must include the inner sqrt argument and the 5/√(10t+12) multiplier.
- Option 2: '-  sin()' is essentially the same as Option 1 but with extra space; still missing the inner argument and the coefficient, so it is incorrect.
- Option 3: 'sin()' omits the negative sign and the chain rule factors entirely; also lacks the inner argument, so incorrect.
- Option 4: '-sin' is incomplete because it does not include the necessary inner function argument or the chain-rule multiplier; it cannot represent the derivative of cos(√(10t+12)).

In summary, none of the provided options match the correct derivative form, which should be -(5/√(10t+12)) sin(√(10t+12)). The key ideas are applying the derivative of cosine, the chain rule, and the derivative of a square root with respect to t.

Find dy/dt. y = cos() 单项选择题

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