{
"query": {
"display": "derivative of $$2e^{2x}$$",
"symbolab_question": "PRE_CALC#derivative 2e^{2x}"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Derivatives",
"subTopic": "Derivatives",
"default": "4e^{2x}",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2e^{2x}\\right)=4e^{2x}$$",
"input": "\\frac{d}{dx}\\left(2e^{2x}\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{d}{dx}\\left(e^{2x}\\right)"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{2x}\\frac{d}{dx}\\left(2x\\right)$$",
"input": "\\frac{d}{dx}\\left(e^{2x}\\right)",
"result": "=e^{2x}\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=2x$$"
],
"result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(2x\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$",
"input": "\\frac{d}{du}\\left(e^{u}\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw=="
}
},
{
"type": "step",
"result": "=e^{u}\\frac{d}{dx}\\left(2x\\right)"
},
{
"type": "step",
"primary": "Substitute back $$u=2x$$",
"result": "=e^{2x}\\frac{d}{dx}\\left(2x\\right)"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqirdytv+QTEbRhtSnwVrzCTdaV09PMxEKZ9FieghTFwc6p4sNpoW3XPzo2W3Ux6aosjLe8tD9HbrkG8vq6q9jjdcfGhUiomJACUgNhWPIuEnsy9CjWbvIjPRIqnDIePUfC30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$",
"input": "\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$",
"result": "=2\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=2",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw"
}
},
{
"type": "step",
"result": "=2e^{2x}\\cdot\\:2"
},
{
"type": "step",
"primary": "Simplify",
"result": "=4e^{2x}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"practiceLink": "/practice/derivatives-practice",
"practiceTopic": "Derivatives"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=4e^{2x}"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
derivative of
Solution
Solution steps
Take the constant out:
Apply the chain rule:
Simplify
Graph
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Frequently Asked Questions (FAQ)
What is the derivative of 2e^{2x} ?
The derivative of 2e^{2x} is 4e^{2x}What is the first derivative of 2e^{2x} ?
The first derivative of 2e^{2x} is 4e^{2x}