Find the derivative of sin(x)ln(x)cos(x)x^2*-1

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

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Answer

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$

Step by step solution

Problem

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

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

$\frac{d}{dx}\left(-x^2\cos\left(x\right)\right)+\frac{d}{dx}\left(\ln\left(x\right)\sin\left(x\right)\right)$
2

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

$\frac{d}{dx}\left(\ln\left(x\right)\sin\left(x\right)\right)-\frac{d}{dx}\left(x^2\cos\left(x\right)\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\cos\left(x\right)$ and $g=x^2$

$\frac{d}{dx}\left(\ln\left(x\right)\sin\left(x\right)\right)-\left(\frac{d}{dx}\left(x^2\right)\cos\left(x\right)+x^2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)$
4

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

$-\left(\frac{d}{dx}\left(x^2\right)\cos\left(x\right)+x^2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)+\ln\left(x\right)\frac{d}{dx}\left(\sin\left(x\right)\right)$
5

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$-\left(2x\cos\left(x\right)+x^2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)+\ln\left(x\right)\frac{d}{dx}\left(\sin\left(x\right)\right)$
6

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

$-\left(2x\cos\left(x\right)+x^2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$
7

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$
8

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)\frac{d}{dx}\left(x\right)+\ln\left(x\right)\cos\left(x\right)$
9

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

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+1\left(\frac{1}{x}\right)\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$
10

Any expression multiplied by $1$ is equal to itself

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$
11

Using the power rule of logarithms

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)+\ln\left(x^{\cos\left(x\right)}\right)$
12

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$

Answer

$-\left(2x\cos\left(x\right)-x^2\sin\left(x\right)\right)+\frac{1}{x}\sin\left(x\right)+\ln\left(x\right)\cos\left(x\right)$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.31 seconds

Views:

100