Applications of Derivatives
Implicit Differentiation of Mixed Exponential Expression
nta_pyq_2023_apr
Grade 12
Question:
If $2x^y+3y^x=20$, then $\dfrac{dy}{dx}$ at $(2,2)$ is equal to:
-\left(\dfrac{2+\log_e 8}{3+\log_e 4}\right)
-\left(\dfrac{3+\log_e 16}{4+\log_e 8}\right)
-\left(\dfrac{3+\log_e 8}{2+\log_e 4}\right)
-\left(\dfrac{3+\log_e 4}{2+\log_e 8}\right)
Step-by-Step Solution
Key Concept: Differentiate $2x^y+3y^x=20$ implicitly: $\frac{d}{dx}(x^y)=x^y(y'\ln x+y/x)$ and $\frac{d}{dx}(y^x)=y^x(\ln y+xy'/y)$.
At $(2,2)$: $8(y'\ln 2+1)+12(\ln 2+y')=0\Rightarrow y'(8\ln2+12)=-8-12\ln2\Rightarrow y'=-\frac{2+\ln8}{3+\ln4}$.
Correct Answer: 1