derivative of 6x^7-2x^5+2

{ "query": { "display": "$$\\frac{d}{dx}\\left(6x^{7}-2x^{5}+2\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(6x^{7}-2x^{5}+2)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "42x^{6}-10x^{4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(6x^{7}-2x^{5}+2\\right)=42x^{6}-10x^{4}$$", "input": "\\frac{d}{dx}\\left(6x^{7}-2x^{5}+2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(6x^{7}\\right)-\\frac{d}{dx}\\left(2x^{5}\\right)+\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(6x^{7}\\right)=42x^{6}$$", "input": "\\frac{d}{dx}\\left(6x^{7}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dx}\\left(x^{7}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6\\cdot\\:7x^{7-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=42x^{6}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnXWMFUzmsCI75SNsVxF9aaTdaV09PMxEKZ9FieghTFwW6NRIpeiyqqqoY+2j0kEuQGILy5Bid0/5E0fv6iIRCe5QV7agSZLIzF7D9vX0CHvlcVRHA6X51k5B4rmITydq7CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x^{5}\\right)=10x^{4}$$", "input": "\\frac{d}{dx}\\left(2x^{5}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(x^{5}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\cdot\\:5x^{5-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=10x^{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqjYgesMfpOdkXSIyXXf2xSTdaV09PMxEKZ9FieghTFwgfhj7yO4KLK0YR8criE8hfsicDtr1/4SZLlnwrW0smO5QV7agSZLIzF7D9vX0CHvNQ2b+35zm99rixeE6LuSsLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(2\\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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "step", "result": "=42x^{6}-10x^{4}+0" }, { "type": "step", "primary": "Simplify", "result": "=42x^{6}-10x^{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=42x^{6}-10x^{4}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }