Question
If \(y=x\sqrt{2x+5}\) , then y′=
A \(\frac{3x+5}{\sqrt{2x+5}}\)
B \(\frac{1}{\sqrt{2x+5}}\)
C \(\frac{1}{2\sqrt{2x+5}}\)
D \(\frac{6x+10}{2\sqrt{2x+5}}\)
Answer/Explanation
Question
The function f is given by \(f(x)=(2x^{3}+b)g(x)\) , where b is a constant and g is a differentiable function satisfying g(2)=4 and g′(2)=−1. For what value of b is f′(2)=0 ?
A −8
B \(-\frac{56}{3}\)
C −24
D −40
Answer/Explanation
Ans:D
Use the product rule to calculate f′(x), then evaluate at x=2 and set equal to 0, as follows.
\(f'(x)=(6x^{2}+b)g(x)+(2x^{3}+bx)g'(x)\)
Question
The table above gives the values of the differentiable functions f and g and their derivatives at x=2. What is the value of \(\frac{\mathrm{d} }{\mathrm{d} x}(f(x)g(x))\) at x=2 ?
A 6
B 13
C 14
D 20
Answer/Explanation
Ans:C
The product rule is applied and correct values are chosen from the table, as follows.
Question
If \(f(x)=\frac{1}{x}.\cos x\) , then f′(x)=
A \(\frac{\sin x}{x^{2}}\)
B \(\frac{-1-x^{2}\sin x}{x^{2}}\)
C \(\frac{-\cos x-x\sin x}{x^{2}}\)
D \(\frac{-\cos x+x\sin x}{x^{2}}\)
Answer/Explanation
Ans:C
The product rule is applied as follows.
\(f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}(\frac{1}{x}).\cos x+\frac{1}{x}\frac{\mathrm{d} }{\mathrm{d} x}(\cos x)\)