Standard +0.3 This is a straightforward implicit differentiation problem requiring the chain rule and product rule, followed by substitution of a given point. While it involves logarithmic differentiation and algebraic manipulation to isolate dy/dx, it's a standard textbook exercise with no novel insight required. The point (1,0) simplifies the algebra considerably, making it slightly easier than average.
State or imply \(\frac{1+\frac{dy}{dx}}{x+y}\) as the derivative of \(\ln(x+y)\)
B1
State or imply \(6xy + 3x^2\frac{dy}{dx}\) as the derivative of \(3x^2y\)
B1
Substitute \((1,0)\) and solve for \(\frac{dy}{dx}\)
M1
Having the correct form for at least one of the above
Obtain \(\frac{dy}{dx} = \frac{1}{2}\) or \(0.5\)
A1
Alternative Method:
Rewrite as \(x+y = e^{3x^2y}\) and state or imply \(1+\frac{dy}{dx}\) as the derivative of the LHS
B1
State or imply \(\left(6xy + 3x^2\frac{dy}{dx}\right)e^{3x^2y}\) as the derivative of \(e^{3x^2y}\)
B1
Substitute \((1,0)\) and solve for \(\frac{dy}{dx}\)
M1
Having the correct form for at least one of the above
Obtain \(\frac{dy}{dx} = \frac{1}{2}\) or \(0.5\)
A1
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\frac{1+\frac{dy}{dx}}{x+y}$ as the derivative of $\ln(x+y)$ | B1 | |
| State or imply $6xy + 3x^2\frac{dy}{dx}$ as the derivative of $3x^2y$ | B1 | |
| Substitute $(1,0)$ and solve for $\frac{dy}{dx}$ | M1 | Having the correct form for at least one of the above |
| Obtain $\frac{dy}{dx} = \frac{1}{2}$ or $0.5$ | A1 | |
| **Alternative Method:** | | |
| Rewrite as $x+y = e^{3x^2y}$ and state or imply $1+\frac{dy}{dx}$ as the derivative of the LHS | B1 | |
| State or imply $\left(6xy + 3x^2\frac{dy}{dx}\right)e^{3x^2y}$ as the derivative of $e^{3x^2y}$ | B1 | |
| Substitute $(1,0)$ and solve for $\frac{dy}{dx}$ | M1 | Having the correct form for at least one of the above |
| Obtain $\frac{dy}{dx} = \frac{1}{2}$ or $0.5$ | A1 | |
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3 The equation of a curve is $\ln ( x + y ) = 3 x ^ { 2 } y$.\\
Find the gradient of the curve at the point $( 1,0 )$.\\
\hfill \mbox{\textit{CAIE P3 2024 Q3 [4]}}