(\partial)/(\partial x)(r+2r^4x+4x)

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(r+2r^{4}x+4x\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(r+2r^{4}x+4x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "2r^{4}+4", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(r+2r^{4}x+4x\\right)=2r^{4}+4$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(r+2r^{4}x+4x\\right)", "steps": [ { "type": "step", "primary": "Treat $$r\\:$$as a constant" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(r\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(2r^{4}x\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(4x\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(r\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(r\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmyZq360qPZOR9HAdfka6zQlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CaRD1/hdiVsHE87DKdSjUF8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(2r^{4}x\\right)=2r^{4}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(2r^{4}x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2r^{4}\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=2r^{4}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2r^{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmfpFKI8dlCQNsPt0JcovgBeefrcs09/UWyGsYd4fMhzbOwcElXShGEqp/uHygEKBp+p3EWKCgqKITWg4Imm3t6N533CjU/CTGsUsDBkHypHGvjY1whc3EJ0M4kpZXkKwY2EDnjh1IjZMkUMmne4eDep/aHde1lsycrFxB0Dxuu+g==" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(4x\\right)=4$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(4x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=4\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=4", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAn1QeFTyw/mEqj4mXw4FVuAnFjOV6V4e2DrBKqW1EhFu4G3acLZKJb8GFYVF4uaxsAOG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrs2LlSh6vRqCs7PDPBxvQ3GwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=0+2r^{4}+4" }, { "type": "step", "primary": "Simplify", "result": "=2r^{4}+4", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }