3.3: Differentiation Rules
- Page ID
- 143971
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- Given \(f(x) = 14\), find \(f'(x)\).
- Given \(g(x) = x^5\), find \(g'(x)\).
- Given \(h(x) = \pi^3\), find \(h'(x)\).
- Given \(y = 2x^3 + 7\), find \(y'\).
- Given \(f(x) = 3x^2 - x + 4\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = \sqrt{2}x^5 - x^{\pi}\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \dfrac{1}{2}(x^3 + x^2 + x + 1)\), find \(\dfrac{dh}{dx}\).
- Given \(y = \dfrac{2x^3 - 4}{3}\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = 2x^{-3} + 4\), find \(f'(x)\).
- Given \(g(x) = \dfrac{2}{x} + \dfrac{x}{2}\), find \(g'(x)\).
- Given \(h(x) = \sqrt{x} - \dfrac{1}{\sqrt{x}}\), find \(h'(x)\).
- Given \(y = \dfrac{1}{2x^{3/2}} + 2\sqrt{x}\), find \(y'(x)\).
- Given \(f(x) = \sqrt[\leftroot{5} \uproot{2} \scriptstyle 3]{\dfrac{8}{x^2}}\), find \(\dfrac{df}{dx}\).
- Given \(g(t) = (t + 1)^2\), find \(\dfrac{dg}{dt}\).
- Given \(h(t) = -5\left(t^3 - 3t^2 - \dfrac{1}{t}\right)\), find \(\dfrac{dh}{dt}\).
- Given \(x = \dfrac{t^2 + 1}{t}\), find \(\dfrac{dx}{dt}\).
- Given \(f(x) = (x^2 + 2x)(2x^3 + 3x^2 - 4)\), find \(f'(x)\).
- Given \(g(x) = (1 + x + x^2)(x^{-1} + x^{-2})\), find \(g'(x)\).
- Given \(h(x) = \left(\dfrac{2}{x^3} - \dfrac{1}{x^5}\right)(x^2 - 5x + 1)\), find \(h'(x)\).
- Given \(y = \dfrac{1}{x^2 + 1}\), find \(y'\).
- Given \(f(x) = \dfrac{x^2 + \sqrt{x}}{x^2 - 1}\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = 3 - \dfrac{2x^{-2} - 4}{x^4}\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \dfrac{(x^2 + 1)(\sqrt{x} - 2)}{2x + 3}\), find \(\dfrac{dh}{dx}\).
- Given \(y = (1 - \sqrt{x})\left(\dfrac{x}{x + 1}\right)\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = |x|\), find \(f'(x)\). [Hint: Consider first the case where \(x > 0\), then separately consider the case where \(x < 0\). For each case, consider how \(|x|\) might be simplified.]
- Find the equation of the line tangent to the graph of \(g(x) = \dfrac{1}{x}\) at \(x = \dfrac{1}{2}\).
- Find the equation of the line tangent to the graph of \(h(x) = \dfrac{x}{1 + x}\) at \(x = 1\).
- Given \(f(x) = \dfrac{1}{6}x^3 - \dfrac{1}{2}x^2 + x - 4\), find \(f'(x)\) and \(f''(x)\).
- Given \(y = \dfrac{x + 1}{x - 1}\), find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\).