{
"query": {
"display": "$$\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)$$",
"symbolab_question": "DERIVATIVE#\\frac{d}{dx}(\\sin(3)x^{2}\\cos(2x))"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Derivatives",
"subTopic": "Derivatives",
"default": "\\sin(3)(2x\\cos(2x)-2x^{2}\\sin(2x))",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)-2x^{2}\\sin\\left(2x\\right)\\right)$$",
"input": "\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=\\sin\\left(3\\right)\\frac{d}{dx}\\left(x^{2}\\cos\\left(2x\\right)\\right)"
},
{
"type": "step",
"primary": "Aplicar la regla del producto: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$",
"secondary": [
"$$f=x^{2},\\:g=\\cos\\left(2x\\right)$$"
],
"result": "=\\sin\\left(3\\right)\\left(\\frac{d}{dx}\\left(x^{2}\\right)\\cos\\left(2x\\right)+\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)x^{2}\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule",
"practiceTopic": "Product Rule"
}
},
{
"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": "$$\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)=-\\sin\\left(2x\\right)\\cdot\\:2$$",
"input": "\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)",
"steps": [
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)",
"result": "=-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\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": [
"$$f=\\cos\\left(u\\right),\\:\\:u=2x$$"
],
"result": "=\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$",
"input": "\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuOCaaVOQ/y0RGnrcxaPJLr8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zBTby8v9dqkicCwl97RZujimcxh5GfxfsNed5mphvPA8XbuNpg35JEo1Mz1bwBSzW9DMulhoXRrVM26A7dfZaj8="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\frac{d}{dx}\\left(2x\\right)"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=2x$$",
"result": "=-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmg4HtP7a1zZWb8Xc4KE0TqQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEybd4WCnwVDNznFKnB/ZRY2yOGqGcv9anBrVhDS9/WspXULslwBmvdshhYKD65r/iE5Rf3frdgbF5+n2Ksb+s3/X507CbTwUTXt5dzrekFYTr2fPHgwoegH4v8fSGorSAo="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$",
"input": "\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$",
"result": "=2\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw"
}
},
{
"type": "step",
"result": "=-\\sin\\left(2x\\right)\\cdot\\:2"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)+\\left(-\\sin\\left(2x\\right)\\cdot\\:2\\right)x^{2}\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)-2x^{2}\\sin\\left(2x\\right)\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=\\sin(3)(2x\\cos(2x)-2x^{2}\\sin(2x))"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
Solución
Pasos de solución
Sacar la constante:
Aplicar la regla del producto:
Simplificar