AP Calculus AB: 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple – Exam Style questions with Answer- MCQ

Question

The graph of the function f, consisting of two line segments,is shown in the figure above. Let g be the function given by g(x)=2x+1,and let h be the fu.nction given by h(x)=f(g(x)).What is the value of \(h{}'(1)\)?

A -4

B -2

C 4

D 6

E nonexistent

▶️Answer/Explanation

Ans:A

Question

\(\frac{\mathrm{d} }{\mathrm{d} x}(\frac{1}{x^{3}}-\frac{1}{x+x^{2}})\) at x=-1 is

A -6

B -4

C 0

 D 2

E 6 

▶️Answer/Explanation

Ans:B

Question

\(y=\frac{1}{2}x^{\frac{4}{5}}-\frac{3}{x^{5}}\) then \(\frac{\mathrm{d} y}{\mathrm{d} x}\)=?

A \(\frac{2}{5x^{\frac{1}{5}}}+\frac{15}{x^{6}}\)

B \(\frac{2}{5x^{\frac{1}{5}}}+\frac{15}{x^{4}}\)

C \(\frac{2}{5x^{\frac{1}{5}}}-\frac{3}{5x^{4}}\)

D \(\frac{2x^{\frac{1}{5}}}{5}+\frac{15}{x^{6}}\)

E \(\frac{2x^{\frac{1}{5}}}{5}-\frac{3}{5x^{4}}\)

▶️Answer/Explanation

Ans:A

Question

Let 

f

be the function defined by\(f(x)=bx^{2}-13bx+b^{2}+\frac{1}{x^{2}}\) where b is a nonzero constant. Which of the following is an expression for \(f{}'(x)\) ,the derivative of f

(x)

 ?

A \(2bx+3b-\frac{2}{x}\)

B \(2bx+3b-\frac{2}{x^{2}}\)

C \(2bx+3b+b^{2}-\frac{2}{x^{3}}\)

D \(x^{2}+3x+b\)

▶️Answer/Explanation

Ans:B

This question involves using the basic power rule for differentiation with respect to the independent variable x while treating the parameter b  as a constant, as follows.

\(f(x)=\frac{\mathrm{d} }{\mathrm{d} x}(bx^{2}+3bx+b^{2}+x^{-2})\)

\(=2bx+3b+0+(-2)x^{-3}\)

\(=2bx+3b-\frac{2}{x^{3}}\)

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