{
"query": {
"display": "derivada de $$f\\left(x\\right)=\\left(x^{2}+3x-2\\right)^{4}$$",
"symbolab_question": "PRE_CALC#derivative f(x)=(x^{2}+3x-2)^{4}"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Derivatives",
"subTopic": "Derivatives",
"default": "4(x^{2}+3x-2)^{3}(2x+3)",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\left(x^{2}+3x-2\\right)^{4}\\right)=4\\left(x^{2}+3x-2\\right)^{3}\\left(2x+3\\right)$$",
"input": "\\frac{d}{dx}\\left(\\left(x^{2}+3x-2\\right)^{4}\\right)",
"steps": [
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}4\\left(x^{2}+3x-2\\right)^{3}\\frac{d}{dx}\\left(x^{2}+3x-2\\right)$$",
"input": "\\frac{d}{dx}\\left(\\left(x^{2}+3x-2\\right)^{4}\\right)",
"result": "=4\\left(x^{2}+3x-2\\right)^{3}\\frac{d}{dx}\\left(x^{2}+3x-2\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=u^{4},\\:\\:u=\\left(x^{2}+3x-2\\right)$$"
],
"result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dx}\\left(\\left(x^{2}+3x-2\\right)\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$",
"input": "\\frac{d}{du}\\left(u^{4}\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la potencia: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=4u^{4-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "step",
"primary": "Simplificar",
"result": "=4u^{3}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q=="
}
},
{
"type": "step",
"result": "=4u^{3}\\frac{d}{dx}\\left(\\left(x^{2}+3x-2\\right)\\right)"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=\\left(x^{2}+3x-2\\right)$$",
"result": "=4\\left(x^{2}+3x-2\\right)^{3}\\frac{d}{dx}\\left(x^{2}+3x-2\\right)"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiPqsFTZVlgo6mKSv73kSaxkq5kTzwFJX3AJhmz7k4Zq1FQ1qG6fJuWh1wKHeWVTZR/eiQ2bs0wqnHz5TvUcL8H7daJ8jmBnO+OHdYDanf0ivXGPCLzNfsqrXkK80YkZidUtpJCOa633rdG+IoZ5jH3QVhzn4X5g9KhEe17S+1HXTeQKHeh69S6dnv9vSoUoFObNCXWr2TfEdFt2DIOP3ZXMpX78v1Lm/UZh8oEbe0OLJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(x^{2}+3x-2\\right)=2x+3$$",
"input": "\\frac{d}{dx}\\left(x^{2}+3x-2\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(3x\\right)-\\frac{d}{dx}\\left(2\\right)"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$",
"input": "\\frac{d}{dx}\\left(x^{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la potencia: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=2x^{2-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2x",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(3x\\right)=3$$",
"input": "\\frac{d}{dx}\\left(3x\\right)",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=3\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$",
"result": "=3\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=3",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsUnAVaDXCLQOuynYB+k3fTZGku9zFkxwe1dTH8vycb9BbqPJ4kl+ElAajU+EBTcW1NbbqpyK7JQEZdATEJR51i3CwF9WgU8/+rFW242YnYD"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2\\right)=0$$",
"input": "\\frac{d}{dx}\\left(2\\right)",
"steps": [
{
"type": "step",
"primary": "Derivada de una constante: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe"
}
},
{
"type": "step",
"result": "=2x+3-0"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2x+3",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=4\\left(x^{2}+3x-2\\right)^{3}\\left(2x+3\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"practiceLink": "/practice/derivatives-practice",
"practiceTopic": "Derivatives"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=4(x^{2}+3x-2)^{3}(2x+3)"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
derivada de
Solución
Pasos de solución
Aplicar la regla de la cadena: