250+ TOP MCQs on Derivatives & Answers | Class 11 Maths

Mathematics Multiple Choice Questions on “Derivatives”.

1. Find the derivative of e^x2.
a) e^x2
b) 2x
c) 2e^x2
d) 2xe^x2
Answer: d
Clarification: We apply chain rule. First we differentiate x².
(frac{d}{dx}) (x²) = 2x
Now, we know that (frac{d}{dx}) (e^x) = e^x
We differentiate e^x2 in the same manner and then multiply with the derivative of x²
(frac{d}{dx}) (e^x2) = 2xe^x2

2. What is the value of (frac{d}{dx}) (sin⁡ x tan⁡ x)?
a) sin⁡ x + tan⁡ x sec⁡ x
b) cos⁡ x + tan⁡ x sec⁡ x
c) sin⁡ x + tan⁡ x
d) sin⁡ x + tan⁡ x sec²⁡x
Answer: a
Clarification: We follow product rule (frac{d}{dx}) (f.g) = g.(frac{d}{dx}) (f) + f.(frac{d}{dx}) (g)
Here, f = sin⁡ x and g = tan⁡ x
(frac{d}{dx}) (sin⁡ x tan⁡ x) = cos⁡ x tan⁡ x + sec²⁡ x sinx
(frac{d}{dx}) (sin⁡ x tan⁡ x) = sin⁡ x + tan⁡ x sec⁡ x

3. What is the value of (frac{d}{dx}) (sin⁡ x³ cos⁡ x²)?
a) 3x² cos x² cos⁡ x³ + 2x sin⁡ x³ sin x²
b) 3x² cos⁡ x² cos⁡ x³ – 2x sin⁡ x³ sin x²
c) 2x cos x² cos⁡ x³ – 2x sin⁡ x³ sin x²
d) 2x cos x² cos⁡ x³ + 3x² sin⁡ x³ sin x²
Answer: b
Clarification: We follow product rule(frac{d}{dx}) (f.g)= g.(frac{d}{dx}) (f)+ f.(frac{dy}{dx}) (g)
Here f = sin⁡ x³ and g = cos⁡ x²
(frac{d}{dx}) (f) = 3x² cos⁡ x³
(frac{d}{dx}) (g) = -2x sin x²
We now substitute this in our main equation,
= cos⁡ x².3x² cos⁡ x³ + sin⁡ x³.(-2x sin x²)
= 3x² cos x² cos⁡ x³ – 2x sin⁡ x³ sin x²

4. What is the value of (frac{d}{dx}(frac{a^x}{e^x}))?
a) (frac{a^x (ln ,⁡a-a^x)}{e^x})
b) (frac{a^x (ln, ⁡a-e^x)}{e^x})
c) (frac{a^x (ln, ⁡a-1)}{e^x})
d) (frac{a^x (ln, ⁡a-1)}{(e^x)(e^x)})
Answer: c
Clarification: Using quotient rule, we know that, (frac{d}{dx} (frac{f}{g}) = frac{g.frac{d}{dx} (f) – f.frac{d}{dx}(g)}{g^2})
Here, f = a^x and g = e^x
(frac{d}{dx} (frac{a^x}{e^x}) = frac{e^x.frac{d}{dx}(a^x)-a^x.frac{d}{dx}(e^x)}{(e^x)(e^x)})
(frac{d}{dx} (frac{a^x}{e^x}) = frac{e^x a^x ln⁡, a-a^x e^x}{(e^x)(e^x)})
(frac{d}{dx} (frac{a^x}{e^x}) = frac{a^x (ln⁡ ,a-1)}{e^x})

5. What is the value of ln ⁡(frac{3}{x})?
a) ⁡(frac{2}{x^3})
b) ⁡(frac{-3}{x^3})
c) ⁡(frac{-9}{x^3})
d) ⁡(frac{9}{x^3})
Answer: c
Clarification: We use chain rule to find (frac{d}{dx})(ln ⁡(frac{3}{x})).
(frac{d}{dx}(ln frac{3}{x})) = (frac{1}{frac{3}{x}} frac{d}{dx}(frac{3}{x}))
(frac{d}{dx}(ln frac{3}{x})) = (frac{3}{x}(-frac{3}{x^2}))
(frac{d}{dx}(ln frac{3}{x})) = (frac{-9}{x^3})

6. The derivative of ln⁡ e^x = 1. Is the statement true or false?
a) True
b) False
Answer: a
Clarification: We know that ln⁡ e^x = x
(frac{d}{dx}) (x) = 1

7. What is the value of (frac{d}{dx})(e^x sinx + e^x cos ⁡x)?
a) 0
b) 2 cos⁡x
c) 2e^x.sin ⁡x
d) 2e^x.cos⁡ x
Answer: d
Clarification: We need to use product rule in both the terms to get the answer.
(frac{d}{dx}) (f.g) = g.(frac{d}{dx}) (f)+ f.(frac{dy}{dx}) (g)
(frac{d}{dx}) (e^x sin x + e^x cos ⁡x) = (e^x.(frac{d}{dx}) (sin⁡ x) + sin ⁡x.(frac{d}{dx}) (e^x)) + (e^x.(frac{d}{dx}) (cos ⁡x) + cos⁡ x.(frac{d}{dx}) (e^x))
(frac{d}{dx}) (e^x sin x + e^x cos ⁡x) =(e^x.cos⁡ x + sin ⁡x . e^x) + (e^x.(-sin ⁡x) + cos ⁡x.e^x)
(frac{d}{dx}) (e^x sin x + e^x cos ⁡x) = e^x.cos⁡ x + sin⁡ x . e^x – e^x.sin⁡ x + cos ⁡x.e^x
(frac{d}{dx}) (e^x sin x + e^x cos ⁡x) = 2e^x.cos⁡ x

8. What is the value of (frac{d}{dx}) (e^x tan x) at x = 0?
a) 0
b) 1
c) -1
d) 2
Answer: b
Clarification: We need to use product rule in both the terms to get the answer.
(frac{d}{dx}) (f.g) = g.(frac{d}{dx}) (f)+ f.(frac{dy}{dx}) (g)
Here f = e^x and g = tan ⁡x
(frac{d}{dx}) (e^x tan x) = tan⁡ x.(frac{d}{dx}) (e^x) + e^x.(frac{d}{dx}) (tan ⁡x)
(frac{d}{dx}) (e^x tan x) = tan⁡ x.e^x + e^x . sec²⁡x
At x = 0 we get,
= tan ⁡0.e⁰ + e⁰.sec²⁡0
= 0.(1) + 1.(1)
= 1

9. What is the value of (frac{d}{dx})(cos²⁡ x tan⁡ x) at x = 1?
a) -1
b) 0
c) -2
d) 1