Applications of Derivatives
Simplification Before Differentiation
nta_pyq_2024_jan
Grade 12
Question:
If $y=\dfrac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\dfrac{1}{15}(3\cos^2x-5)\cos^3x$, then $96y'\left(\dfrac{\pi}{6}\right)$ is equal to:
Step-by-Step Solution
Key Concept: Simplify the algebraic fraction first: factor numerator and denominator to reduce it to a polynomial. Then differentiate the trigonometric part using chain rule. Evaluate the total derivative at $x=\pi/6$.
Simplify: $y=(\sqrt{x}+1)(\sqrt{x}-1)+\frac{1}{5}\cos^5x-\frac{1}{3}\cos^3x=x-1+\frac{1}{5}\cos^5x-\frac{1}{3}\cos^3x$. $y'=1-\cos^4x\sin x+\cos^2x\sin x$. At $x=\pi/6$: $y'=1-\frac{9}{16}\cdot\frac{1}{2}+\frac{3}{4}\cdot\frac{1}{2}=\frac{32-9+12}{32}=\frac{35}{32}$. $96y'(\pi/6)=96\cdot\frac{35}{32}=105$.
Correct Answer: 105