Consider the function
𝑓
(
𝑥
,
𝑦
)
=
sin
⁡
(
5
𝑥
𝑦
)
. What is
𝑓
𝑦
𝑥
?

Question

Consider the function 
𝑓
(
𝑥
,
𝑦
)
=
sin
⁡
(
5
𝑥
𝑦
)
. What is 
𝑓
𝑦
𝑥
?

BlackTom AI · Accepted Answer

We start by restating the problem: given f(x,y) = sin(5xy), determine the mixed partial derivative f_yx, i.e., the derivative first with respect to y and then with respect to x.
Option 1: 25 y^3 sin(5xy). This expression involves only a sine factor with a cubic power of y and lacks any cosine term or a product with x; it does not match the structure of a derivative of sin(5xy) with respect to y and then x. The original function depends on xy inside the sine, so derivatives typically bring down cosine or sine factors multiplied by powers of x or y; this option is missing those essential components.
Option 2: −25 x y^4 sin(5xy). Here we see a sine factor sin(5xy) multiplied by a polynomial in x and y, including a negative sign and a y^4 term. However, taking successive derivatives of sin(5xy) with respect to y and then x does not produce a sin term with y^4 and no cosine factor, because the derivatives of sin(5xy) generate cos(5xy) terms and factors that are linear in y or x from the chain rule. This option also fails to reflect the correct cosine contribution.
Option 3: 25 x y^3 sin(5xy) − 25 y^2 cos(5xy). This expression contains both a sin term and a cos term, which is plausible for a mixed partial derivative of a sine with an inner linear term 5xy. Yet the coefficients and the exact powers (x y^3 for the sin part and y^2 for the cos part) do not align with the actual computation using the chain rule. In particular, a mixed derivative of sin(5xy) with respect to y then x yields a structure involving cos(5xy) with a factor proportional to y, and a cosine term would come with a coefficient that reflects the chain rule action on x and y; the given powers and combination do not match that
logic.
Option 4: 25 y^2 sin(5xy) − 25 x y^3 cos(5xy). This option features a sin term and a cos term, with a pattern where the cos part multiplies x y^3. While the presence of both sin and cos is characteristic of mixed derivatives of sin(5xy), the specific coefficients and exponents do not correspond to the standard differentiation steps: differentiating sin(5xy) first with respect to y yields a cos(5xy) term times 5x, and subsequent differentiation with respect to x introduces another factor of 5y and an additional cosine or sine transformation according to the product rule, leading to particular powers and signs that differ from this option.

Key point across all options: the actual mixed partial derivative f_yx for f(x,y) = sin(5xy) is obtained by first computing f_y = ∂/∂y sin(5xy) = 5x cos(5xy), then differentiating with respect to x: f_yx = ∂/∂x [5x cos(5xy)] = 5 cos(5xy) + 5x · (−sin(5xy)) · ∂(5xy)/∂x. Since ∂(5xy)/∂x = 5y, this becomes f_yx = 5 cos(5xy) − 25xy sin(5xy).

At this point, note that none of the provided answer options exactly matches the correct expression f_yx = 5 cos(5xy) − 25xy sin(5xy). The discrepancy indicates a mismatch between the computed result and all listed choices. Any choice that omits the cos(5xy) term or misplaces the factors of x and y is inconsistent with the chain rule application for this inner function 5xy inside the sine.

In summary, all given options fail to reproduce the correct mixed partial derivative structure, which should contain both a cos(5xy) term and a sin(5xy) term with coefficients 5 and −25xy, respectively. The absence or incorrect arrangement of these components in each option explains why they do not match the proper derivative.

Consider the function 𝑓 ( 𝑥 , 𝑦 ) = sin ⁡ ( 5 𝑥 𝑦 ) . What is 𝑓 𝑦 𝑥 ?Single choice

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