Exam (elaborations) CALCULUS

CALCULUS AND DIFFERENCIATION QUESTIONS
AND ANSWERS

Basic Differentiation
1. Q: Differentiate f(x)=5x3−2x+4f(x) = 5x^3 - 2x + 4.
A:
f′(x)=ddx(5x3−2x+4)f'(x) = \frac{d}{dx}(5x^3 - 2x + 4)
=15x2−2= 15x^2 - 2
2. Q: Differentiate f(x)=3sin⁡(x)+4cos⁡(x)f(x) = 3\sin(x) + 4\cos(x).
A:
f′(x)=ddx(3sin⁡(x)+4cos⁡(x))f'(x) = \frac{d}{dx}(3\sin(x) + 4\cos(x))
=3cos⁡(x)−4sin⁡(x)= 3\cos(x) - 4\sin(x)
3. Q: Differentiate f(x)=ex2f(x) = e^{x^2}.
A:
f′(x)=ddx(ex2)f'(x) = \frac{d}{dx}(e^{x^2})
Using the chain rule,
f′(x)=2xex2f'(x) = 2xe^{x^2}
4. Q: Differentiate f(x)=ln⁡(x2+1)f(x) = \ln(x^2 + 1).
A:
f′(x)=ddx(ln⁡(x2+1))f'(x) = \frac{d}{dx}(\ln(x^2 + 1))
Using the chain rule,
f′(x)=2xx2+1f'(x) = \frac{2x}{x^2 + 1}
5. Q: Differentiate f(x)=tan⁡(x)f(x) = \tan(x).
A:
f′(x)=sec⁡2(x)f'(x) = \sec^2(x)


Product and Quotient Rule
6. Q: Differentiate f(x)=x2sin⁡(x)f(x) = x^2 \sin(x).
A:
Using the product rule,
f′(x)=2xsin⁡(x)+x2cos⁡(x)f'(x) = 2x\sin(x) + x^2\cos(x)
7. Q: Differentiate f(x)=x3cos⁡(x)f(x) = \frac{x^3}{\cos(x)}.
A:
Using the quotient rule,
f′(x)=(3x2)(cos⁡(x))−x3(−sin⁡(x))cos⁡2(x)f'(x) = \frac{(3x^2)(\cos(x)) -
x^3(-\sin(x))}{\cos^2(x)}
f′(x)=3x2cos⁡(x)+x3sin⁡(x)cos⁡2(x)f'(x) = \frac{3x^2\cos(x) + x^3\
sin(x)}{\cos^2(x)}