Calculus I Final Exam 2026 | Derivatives & Integrals Practice Pack

Calculus I Final Exam 2026 | Derivatives & Integrals Practice
Pack

Section A: Derivatives (Questions 1–10)



1. Differentiate:



F(x)=3x^4-5x^3+2x-7



Solution



Using the power rule:



\frac{d}{dx}(x^n)=nx^{n-1}



F’(x)=12x^3-15x^2+2




2. Differentiate:



Y=\frac{2x^2+3x-1}{x}



Solution



First simplify:

,Y=2x+3-\frac{1}{x}



Differentiate:



Y’=2+\frac{1}{x^2}




3. Find the derivative of:



F(x)=\sqrt{x}+5x^{-2}



Solution



F(x)=x^{1/2}+5x^{-2}



F’(x)=\frac12 x^{-1/2}-10x^{-3}



F’(x)=\frac{1}{2\sqrt{x}}-\frac{10}{x^3}




4. Differentiate using product rule:



Y=(x^2+1)(x^3-2)



Solution

,Product Rule:



(uv)’=u’v+uv’



Y’=2x(x^3-2)+(x^2+1)(3x^2)



Simplify:



Y’=2x^4-4x+3x^4+3x^2



Y’=5x^4+3x^2-4x




5. Differentiate using quotient rule:



Y=\frac{x^2+1}{x-3}



Solution



Quotient Rule:



\left(\frac{u}{v}\right)’=\frac{vu’-uv’}{v^2}



Y’=\frac{(x-3)(2x)-(x^2+1)(1)}{(x-3)^2}



Simplify:

, Y’=\frac{2x^2-6x-x^2-1}{(x-3)^2}



Y’=\frac{x^2-6x-1}{(x-3)^2}




6. Differentiate:



Y=\sin x + \cos x



Solution



Y’=\cos x-\sin x




7. Find:



\frac{d}{dx}(e^{2x})



Solution



Chain rule:



\frac{d}{dx}(e^{u})=e^u\cdot u’



\frac{d}{dx}(e^{2x})=2e^{2x}