Try It Yourself#
Exercise 1#
Find the derivative of the function \(f(x) = 5x^2 - 2x\) using the limit definition of the derivative.
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Answer: \(10x - 2\)
Exercise 2#
Evaluate \(f'(x)\) for \(f(x)= 3x^2 + 5x^{-2} + 7\sqrt{x}\) using the basic rules of differentiation.
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Answer: \(6x - 10x^{-3} + \frac{7}{2}x^{-1/2}\)
Exercise 3#
Evaluate \(f'(x)\) for
(Avoid using the quotient rule by rewriting each term so that it involves multplying by a negative power of \(x\).)
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Answer: \(-49x^{-8} - 25x^{-6} - 9x^{-4}\)
Exercise 4#
Find the equation of the tangent line of
(Avoid using the quotient rule by dividing each term of the numerator by \(x^2\) and then simplifying.)
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Answer: \(y = -\frac{9}{64}(x-4) + \frac{11}{16}\)
Exercise 5#
Evaluate \(f'(x)\) for
(Simplify \((x-2)(x+2)\) and then apply the product rule.)
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Answer: \(5x^4 - 12x^2 + 12x\)
Exercise 6#
Evaluate \(f'(x)\) for
(Simplify the function before computing the derivative.)
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Answer: \(\frac{1}{2\sqrt{x-1}}\)
Exercise 7#
Evaluate \(f'(x)\) for
(Rewrite the function before differentiating in order to avoid using the quotient rule.)
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Answer: \(-\frac{102}{(3x+4)^3}\)
Exercise 8#
Evaluate \(f'(x)\) for \(f(x)=5x^4(x^3+6)^{7}\) and simplify your answer.
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Answer: \(5x^3(x^3+6)^6(25x^3 + 24)\)
Exercise 9#
Evaluate \(f'(x)\) for \(f(x)=\dfrac{(x^2+4)^5}{(x^6+ 1)^3}\) and simplify your answer.
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Answer: \(\frac{2x(x^2+4)^4(-4x^6-36x^4+5)}{(x^6+1)^4}\)
Exercise 10#
Evaluate
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Answer: \(5\)
Exercise 11#
Draw the graph of a function \(f(x)\) that is continuous at \(x=a\) but not differentiable at \(x=a\). Explain why the function is continuous but not differentiable at this point.
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Answer: Consider a graph with a corner or a vertical tangent line