derivative of f(x)=(x^2-7)(x^2+6)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\left(x^{2}-7\\right)\\left(x^{2}+6\\right)$$", "symbolab_question": "PRE_CALC#derivative f(x)=(x^{2}-7)(x^{2}+6)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "4x^{3}-2x", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\left(x^{2}-7\\right)\\left(x^{2}+6\\right)\\right)=4x^{3}-2x$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}-7\\right)\\left(x^{2}+6\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x^{2}-7,\\:g=x^{2}+6$$" ], "result": "=\\frac{d}{dx}\\left(x^{2}-7\\right)\\left(x^{2}+6\\right)+\\frac{d}{dx}\\left(x^{2}+6\\right)\\left(x^{2}-7\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-7\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}-7\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)-\\frac{d}{dx}\\left(7\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(7\\right)=0$$", "input": "\\frac{d}{dx}\\left(7\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr/PPZDkjJg1JcijqZvGH+JJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuldgLTbPmvou0JbfidNFAp" } }, { "type": "step", "result": "=2x-0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+6\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+6\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(6\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(6\\right)=0$$", "input": "\\frac{d}{dx}\\left(6\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg8p7Gq8hcikAAMclWLaxZJJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtyoiJomQLyoKTDK4FJPEzd" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2x\\left(x^{2}+6\\right)+2x\\left(x^{2}-7\\right)" }, { "type": "interim", "title": "Simplify $$2x\\left(x^{2}+6\\right)+2x\\left(x^{2}-7\\right):{\\quad}4x^{3}-2x$$", "input": "2x\\left(x^{2}+6\\right)+2x\\left(x^{2}-7\\right)", "result": "=4x^{3}-2x", "steps": [ { "type": "interim", "title": "Expand $$2x\\left(x^{2}+6\\right):{\\quad}2x^{3}+12x$$", "input": "2x\\left(x^{2}+6\\right)", "result": "=2x^{3}+12x+2x\\left(x^{2}-7\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2x,\\:b=x^{2},\\:c=6$$" ], "result": "=2xx^{2}+2x\\cdot\\:6", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2x^{2}x+2\\cdot\\:6x" }, { "type": "interim", "title": "Simplify $$2x^{2}x+2\\cdot\\:6x:{\\quad}2x^{3}+12x$$", "input": "2x^{2}x+2\\cdot\\:6x", "result": "=2x^{3}+12x", "steps": [ { "type": "interim", "title": "$$2x^{2}x=2x^{3}$$", "input": "2x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=2x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70thjYFL6CU6+q37zoPyD/wOfOVs9mPIqDLV5QIWwt3mwB/QJ3d78LroQvy/1JTpnx06diEhTBX3c/BKR15lU3GMU8hJEL7k4ZGKY8HFIubc=" } }, { "type": "interim", "title": "$$2\\cdot\\:6x=12x$$", "input": "2\\cdot\\:6x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:6=12$$", "result": "=12x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7APyI3K+HoU90nE0np8T5qyAn9lkDfZkicUGkO3EF+Io7TJgXBzmRSM2ZmvJL/c8RlEd8bz72O1EcTNHc6v8UnYwWl+00m8w1D2Un4YDQITw=" } }, { "type": "step", "result": "=2x^{3}+12x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+BkGTB9WwsI3MXtKAYkYEN6GQqufR6tr2vPxOUv7H++pckVsfuWTwR902mB6Yi0v1eHa/FkkCyi6OGmgN4EUMh7MWKURN+43KCOzRc+RXaHcM6aqrMwMhpstaLXHeaOnsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Expand $$2x\\left(x^{2}-7\\right):{\\quad}2x^{3}-14x$$", "input": "2x\\left(x^{2}-7\\right)", "result": "=2x^{3}+12x+2x^{3}-14x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2x,\\:b=x^{2},\\:c=7$$" ], "result": "=2xx^{2}-2x\\cdot\\:7", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2x^{2}x-2\\cdot\\:7x" }, { "type": "interim", "title": "Simplify $$2x^{2}x-2\\cdot\\:7x:{\\quad}2x^{3}-14x$$", "input": "2x^{2}x-2\\cdot\\:7x", "result": "=2x^{3}-14x", "steps": [ { "type": "interim", "title": "$$2x^{2}x=2x^{3}$$", "input": "2x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=2x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70thjYFL6CU6+q37zoPyD/wOfOVs9mPIqDLV5QIWwt3mwB/QJ3d78LroQvy/1JTpnx06diEhTBX3c/BKR15lU3GMU8hJEL7k4ZGKY8HFIubc=" } }, { "type": "interim", "title": "$$2\\cdot\\:7x=14x$$", "input": "2\\cdot\\:7x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:7=14$$", "result": "=14x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WAm0tvklu5NZiY2/8RQZbiAn9lkDfZkicUGkO3EF+IqBOXTBy1T3bcHqcGrdVz8zlEd8bz72O1EcTNHc6v8UndKVNK5qAKdrD9mO7eJj8bQ=" } }, { "type": "step", "result": "=2x^{3}-14x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IZLUcuICtYq481ZgW87jaN6GQqufR6tr2vPxOUv7H++pckVsfuWTwR902mB6Yi0vRpmTJVISwPkZ82KF4/qcAR7MWKURN+43KCOzRc+RXaHN99iSQvkn5+zZYAmZc0QJsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$2x^{3}+12x+2x^{3}-14x:{\\quad}4x^{3}-2x$$", "input": "2x^{3}+12x+2x^{3}-14x", "result": "=4x^{3}-2x", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2x^{3}+2x^{3}+12x-14x" }, { "type": "step", "primary": "Add similar elements: $$2x^{3}+2x^{3}=4x^{3}$$", "result": "=4x^{3}+12x-14x" }, { "type": "step", "primary": "Add similar elements: $$12x-14x=-2x$$", "result": "=4x^{3}-2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/z29886vVYd3TCveq2TDA65+204lTn1IGDXevTwbQNQDnzlbPZjyKgy1eUCFsLd5s066PvJ7kqJdE96gbB7+UmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9OZy+TbQ8/hepPxZf77BGyoa12JbFLvmUIYonJIdo6/M=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=4x^{3}-2x" }, "showViewLarger": true } }, "meta": { "showVerify": true } }