WBJEE · Maths · Differentiation
If \(f(x)=x^{n}, n\) being a non-negative integer, then the values of \(n\) for which
\(f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)\) for all \(\alpha, \beta>0\) is
- A 1
- B 2
- C 0
- D 5
Answer & Solution
Correct Answer
(C) 0
Step-by-step Solution
Detailed explanation
We have, \(\begin{aligned} f(x) &=x^{n} \\ f^{\prime}(x) &=n x^{n-1} \end{aligned}\) Now, \(f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)\) \(\Rightarrow \quad n(\alpha+\beta)^{n-1}=n \alpha^{n-1}+n \beta^{n-1}\) \(\Rightarrow(a+\beta)^{n-1}=a^{n-1}+\beta^{n-1}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- ln a\(\Delta A B C,\) if \(\angle C=90^{\circ}, r\) and \(R\) are the inradius and circumrodius of the ABC respectively, then \(2(r+R)\) is equal toWBJEE 2015 Hard
- Three unequal positive numbers a, b, c are such that a, b, c are in G.P. while \(\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)\) are in A.P. Then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are the lengths of the sides ofWBJEE 2021 Medium
- Each of \(a\) and \(b\) can take values 1 or 2 with equal probability. The probability that the equation \(a x^{2}+b x+1=0\) has real roots, is equal toWBJEE 2013 Easy
- A point is in motion along a hyperbola \(y=\frac{10}{x}\) so that its abscissa \(x\) increases uniformly at a rate of 1 unit per second. Then, the rate of change of its ordinate when the point passes through (5,2)WBJEE 2019 Easy
- \(\int \frac{\sin ^{-1} x}{\sqrt{1-x^2}} d x\) is equal to
where \(c\) is an arbitrary constantWBJEE 2009 Easy - A straight line through the origin \(\mathrm{O}\) meets the parallel lines \(4 \mathrm{x}+2 \mathrm{y}=9\) and \(2 \mathrm{x}+\mathrm{y}+6=0\) at \(\mathrm{P}\) and \(\mathrm{Q}\) respectively. The point O divides the segment \(\mathrm{PQ}\) in the ratioWBJEE 2020 Easy
More PYQs from WBJEE
- A proton is moving with a uniform velocity of \(10^{6} \mathrm{ms}^{-1}\) along the \(Y\) -axis, under the joint action of a magnetic field along \(Z\) -axis and an electric field of magnitude \(2 \times 10^{4} \mathrm{Vm}^{-1}\) along the negative \(X\) -axis. If the electric field is switched off, the proton starts moving in a circle. The radius of the circle is nearly given: \(\frac{e}{m}\) ratio for proton \(=10^{4} \mathrm{Ckg}^{-1}\)WBJEE 2017 Medium
- Let \(f(n)=2^{n+1}, g(n)=1+(n+1)^{2 n}\) for all \(n \in \mathbb{N}\). ThenWBJEE 2022 Medium
- If \(\left(1+x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots+a_{12} x^{12}\), then the value of \(a_2+a_4+a_6+\ldots+a_{12}\) isWBJEE 2025 Medium
- If \(\left(\log _{5} x\right)\left(\log _{x} 3 x\right)\left(\log _{3 x} y\right)=\log _{x} x^{3},\) then \(y\) equalsWBJEE 2017 Easy
- Let \(A\) and \(B\) be two events with \(P\left(A^{C}\right)=0.3\) \(P(B)=0.4 \quad\) and \(\quad P\left(A \cap B^{C}\right)=0.5 . \quad\) Then
\(P\left(B \mid A \cup B^{c}\right)\) is equal toWBJEE 2012 Easy - Five numbers are in HP. The middle term is 1 and the ratio of the second and the fourth terms is 2: 1 . Then, the sum of the first three terms isWBJEE 2013 Medium