The derivative of [math: (2x+5)6] (2x+5)^6 is:

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First, restating the problem helps orient the analysis: we are finding the derivative of (2x+5)^6 with respect to x.
Option a: 12(2x+5)^5. This matches the chain rule: if y = (u)^6 with u = 2x+5, dy/dx = 6(u)^5 * du/dx, and du/dx = 2. Multiplying gives 6*(2x+5)^5*2 = 12(2x+5)^5. This is correct.
Option b: 6(2x+5)^5. This would be the result if we forgot the outer derivative factor of 6, or if we misapplied the inner derivative. It ignores the factor of 2 from derivative of (2x+5), so it underestimates.
Option c: 12x(2x+5)^5. This introduces an extra x factor, which would be appropriate if the inner function were x(2x+5) or if differentiating a product, but here the inner function is simply (2x+5). The presence of x is incorrect.
Option d: 6x(2x+5)^5. Similar to option c, this multiplies by x unnecessarily and reduces the outer coefficient from 12 to 6, which does not align with the chain rule application for (2x+5)^6.
In summary, applying the chain rule correctly requires multiplying by the derivative of the inner function (which is 2) and by the outer exponent factor (6), yielding 6*(2x+5)^5*2 = 12(2x+5)^5. The other options either omit the inner derivative, introduce an extra x, or misplace factors, leading to incorrect expressions.

MUF0092 Mathematics Unit 2-SEM 1-2025 1.5 Derivatives Revision

The derivative of [math: (2x+5)6] (2x+5)^6 is:

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