Math / Calculus

QuestionUse logarithmic differentiation to find the derivative of the function. $\begin{array}{r} y=x^{2} \cos (x) \\ y^{\prime}=\square \end{array}$

Studdy Solution

STEP 1

1. The function $y = x^2 \cos(x)$ involves both polynomial and trigonometric components.
2. Logarithmic differentiation is a technique that can simplify the differentiation of products and quotients of functions.
3. The derivative of the function should be found using the properties of logarithms and the rules of differentiation.

STEP 2

1. Take the natural logarithm of both sides.
2. Differentiate implicitly with respect to $x$ .
3. Solve for $y'$ and simplify.

STEP 3

Take the natural logarithm of both sides of the equation $y = x^2 \cos(x)$ .
$\ln(y) = \ln(x^2 \cos(x))$

STEP 4

Use the properties of logarithms to separate the product inside the logarithm.
$\ln(y) = \ln(x^2) + \ln(\cos(x))$

STEP 5

Simplify the expression using the logarithm power rule.
$\ln(y) = 2\ln(x) + \ln(\cos(x))$

STEP 6

Differentiate both sides of the equation with respect to $x$ implicitly.
$\frac{d}{dx} [\ln(y)] = \frac{d}{dx} [2\ln(x) + \ln(\cos(x))]$

STEP 7

Apply the chain rule to the left side of the equation.
$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} [2\ln(x)] + \frac{d}{dx} [\ln(\cos(x))]$

STEP 8

Differentiate the right side of the equation term by term.
$\frac{1}{y} y' = 2 \frac{1}{x} + \frac{1}{\cos(x)} \left( -\sin(x) \right)$

STEP 9

Simplify the expression on the right side.
$\frac{1}{y} y' = \frac{2}{x} - \tan(x)$

STEP 10

Solve for $y'$ by multiplying both sides by $y$ .
$y' = y \left( \frac{2}{x} - \tan(x) \right)$

STEP 11

Substitute back $y = x^2 \cos(x)$ into the expression.
$y' = x^2 \cos(x) \left( \frac{2}{x} - \tan(x) \right)$

STEP 12

Simplify the final expression.
$y' = x^2 \cos(x) \left( \frac{2}{x} - \frac{\sin(x)}{\cos(x)} \right) = x^2 \cos(x) \left( \frac{2}{x} - \frac{\sin(x)}{\cos(x)} \right)$
$y' = x^2 \cos(x) \left( \frac{2}{x} - \frac{\sin(x)}{\cos(x)} \right) = x^2 \cos(x) \left( \frac{2}{x} - \frac{\sin(x)}{\cos(x)} \right) = x \cos(x) \left(2 - x \tan(x)\right)$
The derivative of the function is: $y' = x \cos(x) (2 - x \tan(x))$

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