Practice questions on computing and understanding the properties of the derivative of a function in calculus[cite: 1]. These questions help you gain a deeper understanding of derivative properties. Each solution is hidden inside a collapsible dropdown so you can practice independently before reviewing the steps.
True or False: If a function is continuous at \(x = a\), then it has a tangent line at \(x = a\).
Answer: False. For example, the function \(f(x) = |x|\) is continuous at \(x = 0\) but has no tangent line at that point.
True or False: The derivative of a function at a given point gives the slope of the tangent line at that point.
Answer: True. This follows directly from the definition of the derivative.
True or False: If \(f'\) is the derivative of \(f\), then the derivative of the inverse of \(f\) is the inverse of \(f'\).
Answer: False. If \(g(x)\) is the inverse of \(f(x)\), its derivative is:
\[ g'(x) = \frac{1}{f'(g(x))}. \]True or False: The derivative of \(\ln(ax)\), where \(a\) is a constant, is \(\frac{1}{x}\).
Answer: True.
True or False: Rolle's theorem is a special case of the Mean Value Theorem.
Answer: True.
If \(f(x) = x^3 - 3x^2 + x\) and \(g\) is the inverse of \(f\), then \(g'(3)\) is equal to:
(A) 10
(B) \(\dfrac{1}{10}\)
(C) 1
(D) None of the above
Answer: (B). Using \(g'(x) = \frac{1}{f'(g(x))}\), first find \(g(3)\) by solving \(f(x) = 3\):
\[ x^3 - 3x^2 + x = 3 \quad \Rightarrow \quad x = 3 \Rightarrow g(3) = 3 \]Then compute \(f'(x) = 3x^2 - 6x + 1\) and evaluate:
\[ f'(g(3)) = 3 \cdot 3^2 - 6 \cdot 3 + 1 = 10 \quad \Rightarrow \quad g'(3) = \frac{1}{10}. \]True or False: The derivative of \(f(x) = a^x\), where \(a\) is a constant, is \(x a^{x-1}\).
Answer: False. Let \(y = a^x\) so that \(\ln y = x \ln a\). Differentiating both sides:
\[ \frac{1}{y} \frac{dy}{dx} = \ln a \quad \Rightarrow \quad \frac{dy}{dx} = y \ln a = a^x \ln a. \]
Calculus questions with answers[cite: 1]
Calculus tutorials and problems[cite: 1].