Math Assignment Class XII Chapter 5 Derivatives
Question 1 Use chain rule to find the derivative of
Question 2. Differentiate the following w.r.t. x
Answer
Solution Hint
Now differentiating w. r. t. x and taking log10 as constant.
Question 3. Differentiate the following w. r. t. x at x = 1
Answer: 2e
Question 4.
, Then prove that : Question 5: Differentiate log(x ex) w. r. t. xlogx.
Answer 5:
Question 6: Differentiate x2 w. r. t. x3. ........... Ans:
2/3x.
Question 7: , Prove that
Solution Hint:
Dividing on both side by
Now differentiating w. r. t. x we get the required answer.
Question 8: Differentiate (xx)x w. r. t. x
Answer
Solution Hint:
Now differentiating w. r. t. x
Question9. Differentiate
x16y9 = (x2 + y)17 w. r. t. x ,
Answer : Question10) Differentiate (logx)x + xlogx, w. r.
t. x.
Answer
Question11
Answer: Question 12:
Then prove that :
Solution 12:
Differentiating both side w. r. t. x we get
Question 13. Find dy/dx at x = 1, y = π/4, If Sin2y + cosxy = k
Answer 13.
Question 14. If x = a(2θ - sin2θ) and y = a(1-cos2θ), find dy/dx, when θ = π/4.
Answer 14: [ 1]
Question 15. Differentiate Log(cosex) w. r. t x
Ans[-ex tanx].
Question 16.
then prove that
Question 17. then prove that
Question 18.
, then find the value of
Question 19: Differentiate the following w. r. t. x
Ans: Question 20. Differentiate the following w. r. t. x
Ans: Question 21. Differentiate: Ans: Question 22.
Find the value of k if the following function is continuous.\[f(x)=\left\{\begin{matrix} \frac{kcosx}{\pi -2x}\; \; ,\; \; x<\frac{\pi }{2}\\\\\frac{3tan2x}{2x-\pi }\; \; ,\; \; x>\frac{\pi }{2} \end{matrix}\right.\; \; \; \; \; \; \; \: \: \: ....\: \: Ans\: \: k=6\]
Question 23.
Find the value of k for which the following function is continuous at x = 0\[f(x)=\left\{\begin{matrix} \frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}\; \; ,\; \; -1\leq x<0\\\\ \frac{2x+1}{x-2}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \:, \: \: \: \: \: \: 0\leq x<1 \end{matrix}\right.\: \: \: \: \: .....\: \: \: \: \: \: Ans\: k=-\frac{1}{2}\]
Question 24:
Show that f(x) = |x - 2| is continuous but not differentiable at x = 2.
Question 25:
Discuss the continuity and differentiability of the following function. \[f(x)=\left\{\begin{matrix} 1-x\: \:\: \: \: \: \: \: ,\: \: x<1\\(1-x)(2-x),\: 1\leq x\leq 2 \\ 3-x\: \: \: \: \: ,\: \: \: \: \: x>2 \end{matrix}\right.\]
Answer 25
f(x) is continuous at x = 1 but not continuous at x = 2
f(x) is differentiable at x = 1.
As f(x) is not continuous at x = 2 so f(x) is not differentiable at x = 2.
Question 27
Check the differentiability of the function
Solution Hint
LHD = RHD ⇒ f(x) is differentiable at x = 2
Question 28
If f(x) = (x + 1)cotx is continuous at x = 0 then find f(0).
Answer = e
Solution: We know that
Now, f(x) = (x+1)cotx
Log[f(x)] = Log[(x+1)cotx ]
Log[f(x)] = cotx Log(x+1)
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