Answer to Question #315309 in Calculus for Niu_bi

Question #315309

ACTIVITY IN BASIC CALCULUS

BASIC RULES IN DERIVATIVE


 Complete the blanks of the given function below with a number (except 0 and 1) to create your own problem and find the derivative of the function. Show your complete solution to each problem.

 

1. f(x) = -4x5 + ______x-4- 2468


2. f (x) =____x-3- _____x1/4 - 12x


3.f(x)= ____ "\\sqrt[4]{x}" 3 - "\\underset{x^6}{=}" + "\\frac{2}{3}" x6

4.f (x) = "\\underset{x^-6}{=}" - ____x2 + "\\sqrt[4]{x}"


 


1
Expert's answer
2022-03-23T19:18:50-0400

1.


"f(x) = -4x^5 + 3x^{-4} - 2468\\\\\n\\begin{aligned}\n f'(x)= & -20 x^{4}-\\frac{12}{x^{5}}\\\\[2mm]\nf'(x) =& -\\frac{20 x^{9}+12}{x^{5}}\n\\end{aligned}"

2.


"f (x) = 6x^{-3} -12x^{1\/4} - 12x\\\\\n\\begin{aligned}\nf'(x) = & \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[-12 x-12 \\sqrt[4]{x}+\\frac{6}{x^{3}}\\right] \\\\\n= & -12 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}[x]-12 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}[\\sqrt[4]{x}]+6 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[\\frac{1}{x^{3}}\\right] \\\\\n= & -12 \\cdot 1-12 \\cdot \\frac{1}{4} x^{\\frac{1}{4}-1}+6(-3) x^{-4} \\\\\nf'(x)= & -\\frac{3}{x^{\\frac{3}{4}}}-\\frac{18}{x^{4}}-12\n\\end{aligned}"

3.


"f(x) = 5\\sqrt[4]{x^3}-\\frac{3}{x^6}+\\frac{2}{3}x^6\\\\\n\\begin{aligned}\nf'(x) =& \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[\\frac{2 x^{6}}{3}+5 x^{\\frac{3}{4}}-\\frac{3}{x^{6}}\\right] \\\\\n= & \\frac{2}{3} \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[x^{6}\\right]+5 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[x^{\\frac{3}{4}}\\right]-3 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[\\frac{1}{x^{6}}\\right] \\\\\n=& \\frac{2 \\cdot 6 x^{5}}{3}+5 \\cdot \\frac{3}{4} x^{\\frac{3}{4}-1}-3(-6) x^{-7} \\\\\nf'(x) = & 4 x^{5}+\\frac{15}{4 \\sqrt[4]{x}}+\\frac{18}{x^{7}}\n\\end{aligned}"

4.


"f(x) = \\frac{7}{x^{-6}}-4x^2 + {x^{1\/4}}\\\\\n\\begin{aligned}\nf'(x) = &\\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[7 x^{6}-4 x^{2}+\\sqrt[4]{x}\\right] \\\\\n= &7 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[x^{6}\\right]-4 \\cdot \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left[x^{2}\\right]+\\frac{\\mathrm{d}}{\\mathrm{d} x}[\\sqrt[4]{x}] \\\\\n=& 7 \\cdot 6 x^{5}-4 \\cdot 2 x+\\frac{1}{4} x^{\\frac{1}{4}-1} \\\\\nf'(x)=& 42 x^{5}-8 x+\\frac{1}{4 x^{\\frac{3}{4}}}\n\\end{aligned}"


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