Find the derivative of e^(2x)xcos(xy)*-1

\frac{d}{dx}\left(e^{2x}-x\cdot \cos\left(xy\right)\right)

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Answer

$2e^{2x}-\left(\cos\left(y\cdot x\right)-x\cdot y\sin\left(y\cdot x\right)\right)$

Step by step solution

Problem

$\frac{d}{dx}\left(e^{2x}-x\cdot \cos\left(xy\right)\right)$
1

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(-x\cos\left(y\cdot x\right)\right)+\frac{d}{dx}\left(e^{2x}\right)$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(e^{2x}\right)-\frac{d}{dx}\left(x\cos\left(y\cdot x\right)\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\cos\left(y\cdot x\right)$

$\frac{d}{dx}\left(e^{2x}\right)-\left(x\frac{d}{dx}\left(\cos\left(y\cdot x\right)\right)+\cos\left(y\cdot x\right)\frac{d}{dx}\left(x\right)\right)$
4

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(e^{2x}\right)-\left(x\frac{d}{dx}\left(\cos\left(y\cdot x\right)\right)+1\cos\left(y\cdot x\right)\right)$
5

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{d}{dx}\left(e^{2x}\right)-\left(1\cos\left(y\cdot x\right)-x\frac{d}{dx}\left(y\cdot x\right)\sin\left(y\cdot x\right)\right)$
6

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(e^{2x}\right)-\left(1\cos\left(y\cdot x\right)-x\cdot y\sin\left(y\cdot x\right)\frac{d}{dx}\left(x\right)\right)$
7

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(e^{2x}\right)-\left(1\cos\left(y\cdot x\right)-1\cdot 1x\cdot y\sin\left(y\cdot x\right)\right)$
8

Applying the derivative of the exponential function

$1\frac{d}{dx}\left(2x\right)e^{2x}-\left(1\cos\left(y\cdot x\right)-1\cdot 1x\cdot y\sin\left(y\cdot x\right)\right)$
9

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$1\cdot 2e^{2x}\cdot\frac{d}{dx}\left(x\right)-\left(1\cos\left(y\cdot x\right)-1\cdot 1x\cdot y\sin\left(y\cdot x\right)\right)$
10

The derivative of the linear function is equal to $1$

$1\cdot 1\cdot 2e^{2x}-\left(1\cos\left(y\cdot x\right)-1\cdot 1x\cdot y\sin\left(y\cdot x\right)\right)$
11

Multiply $1$ times $2$

$2e^{2x}-\left(1\cos\left(y\cdot x\right)-x\cdot y\sin\left(y\cdot x\right)\right)$
12

Any expression multiplied by $1$ is equal to itself

$2e^{2x}-\left(\cos\left(y\cdot x\right)-x\cdot y\sin\left(y\cdot x\right)\right)$

Answer

$2e^{2x}-\left(\cos\left(y\cdot x\right)-x\cdot y\sin\left(y\cdot x\right)\right)$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.29 seconds

Views:

126