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Fittoniya [83]
3 years ago
8

Find the Derivative. f(x) = -xcos3x

Mathematics
2 answers:
DENIUS [597]3 years ago
3 0

Answer:

\frac{d}{dx}(-xcos3x)=  3x \ sin(3x)- cos(3x)

Step-by-step explanation:

Use the Product Rule for derivatives, which states that:

  • \frac{d}{dx} =[f(x)g(x)]= f(x)g'(x)+f'(x)g(x)

In the function we are given, f(x)=-x \cos3x, we can break it up into two factors: -x is being multiplied by cos3x.

Now, we have the factors:

  • -x \\   $cos 3x

Before using the product rule, let's find the derivative of cos3x using the chain rule and the power rule.

  • \frac{d}{dx}(cos3x)= \frac{d}{dx} (cos3x) \times \frac{d}{dx} (3x) \\ \frac{d}{dx}(cos3x)= (-sin3x) \times 3 \\ \frac{d}{dx} (cos3x) = -3sin(3x)

Now let's apply the product rule to f(x) = -xcos3x.

  • \frac{d}{dx}(-xcos3x)=  (-x)(-3sin3x) + (-1)(cos3x)

Simplify this equation.

  • \frac{d}{dx}(-xcos3x)=  (x)(3sin3x) - (cos3x)

Multiply x and 3 together and remove the parentheses.

  • \frac{d}{dx}(-xcos3x)=  3x \ sin(3x)- cos(3x)

Therefore, this is the derivative of the function f(x)=-xcos3x.

e-lub [12.9K]3 years ago
3 0

Hi there!

\large\boxed{f'(x) =  -cos3x + 3xsin3x}

f(x) = -xcox3x\\\\\text{Use the product rule and chain rule to solve for the derivative:}\\\\\frac{dy}{dx}= -cos3x + (-x * (-sin3x) * 3)\\\\= -cos3x + 3xsin3x

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