Question

Let $y=x^{\sin\frac{\pi}{2}x}$. Then at $x=1$, $\frac{dy}{dx}$ is equal to

• A. -1
• B. 0
• C. -2
• D. 2
• E. 1

Hint:

Math Tip: When evaluating a derivative at a specific point ($x=1$), you do not need to algebraically isolate $\frac{dy}{dx}$ completely before substituting. Plugging in the numbers immediately after differentiating saves time and reduces algebraic errors!

Answer 1

Answer: (E)

Concept:
Calculus - Logarithmic Differentiation.
When a function has a variable in both the base and the exponent, taking the natural logarithm on both sides simplifies the differentiation process.
Step 1: Apply the natural logarithm to both sides.
Given $y = x^{\sin(\frac{\pi}{2}x)}$. $$ \ln y = \ln\left(x^{\sin(\frac{\pi}{2}x)}\right) $$ Use the logarithm power rule $\ln(a^b) = b\ln a$: $$ \ln y = \sin\left(\frac{\pi}{2}x\right) \cdot \ln x $$
Step 2: Differentiate both sides with respect to x.
Apply implicit differentiation on the left side and the product rule on the right side: $$ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left[ \sin\left(\frac{\pi}{2}x\right) \cdot \ln x \right] $$
Step 3: Execute the product and chain rules.
$$ \frac{1}{y} \frac{dy}{dx} = \sin\left(\frac{\pi}{2}x\right) \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}\left(\sin\left(\frac{\pi}{2}x\right)\right) $$ $$ \frac{1}{y} \frac{dy}{dx} = \sin\left(\frac{\pi}{2}x\right) \cdot \left(\frac{1}{x}\right) + \ln x \cdot \left(\cos\left(\frac{\pi}{2}x\right) \cdot \frac{\pi}{2}\right) $$
Step 4: Calculate the value of y at x = 1.
Before solving for the derivative, find the value of $y$ when $x = 1$: $$ y(1) = 1^{\sin(\frac{\pi}{2} \cdot 1)} = 1^{\sin(\frac{\pi}{2})} $$ Since $\sin(\frac{\pi}{2}) = 1$: $$ y(1) = 1^1 = 1 $$
Step 5: Evaluate the derivative at x = 1.
Substitute $x = 1$ and $y = 1$ into the differentiated equation: $$ \frac{1}{1} \frac{dy}{dx} = \sin\left(\frac{\pi}{2}\right) \cdot \left(\frac{1}{1}\right) + \ln(1) \cdot \left(\cos\left(\frac{\pi}{2}\right) \cdot \frac{\pi}{2}\right) $$ Substitute the known values $\sin(\frac{\pi}{2}) = 1$, $\ln(1) = 0$, and $\cos(\frac{\pi}{2}) = 0$: $$ \frac{dy}{dx} = (1) \cdot (1) + (0) \cdot \left(0 \cdot \frac{\pi}{2}\right) $$ $$ \frac{dy}{dx} = 1 + 0 = 1 $$