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(6 pts) Use logarithmic differentiation to find the derivative of

y=(10x^(2)+1)^(cos x)

$3$ . ( $6$ pts) Use logarithmic differentiation to find the derivative of $\newline$ $y=\left(10 x^{2}+1\right)^{\cos x}$

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Convert between exponential and logarithmic form: all bases

Full solution

Q.

3

. (

6

pts) Use logarithmic differentiation to find the derivative of

\newline

y=\left(10 x^{2}+1\right)^{\cos x}

Apply Logarithmic Differentiation: First, we need to apply logarithmic differentiation. To do this, we take the natural logarithm of both sides of the equation $y=(10x^{2}+1)^{\cos x}$ . $\newline$ We get $\ln(y) = \ln((10x^{2}+1)^{\cos x})$ . $\newline$ Using the property of logarithms that allows us to bring the exponent in front, we get $\ln(y) = \cos(x) \cdot \ln(10x^{2}+1)$ .
Differentiate with Respect to $x$ : Next, we differentiate both sides of the equation with respect to $x$ . The left side becomes the derivative of $\ln(y)$ with respect to $y$ , multiplied by the derivative of $y$ with respect to $x$ (by the chain rule), which is $(1/y) \cdot \frac{dy}{dx}$ . The right side will require the product rule since it is the product of two functions of $x$ , $\cos(x)$ and $\ln(10x^{2}+1)$ . So, we have $x$ $0$ .
Solve for dy/dx: Now we need to solve for $\frac{dy}{dx}$ . We multiply both sides of the equation by $y$ to isolate $\frac{dy}{dx}$ on one side. $\frac{dy}{dx} = y \cdot (-\sin(x) \cdot \ln(10x^{2}+1) + \cos(x) \cdot \left(\frac{1}{10x^{2}+1}\right) \cdot (20x)).$
Substitute Original y: We substitute back the original y from $y=(10x^{2}+1)^{\cos x}$ into the equation. $\newline$ $\frac{dy}{dx} = (10x^{2}+1)^{\cos x} * (-\sin(x) * \ln(10x^{2}+1) + \cos(x) * (\frac{1}{10x^{2}+1}) * (20x))$ .
Simplify the Expression: Finally, we simplify the expression to get the final answer. $\frac{dy}{dx} = (10x^{2}+1)^{\cos x} * (-\sin(x) * \ln(10x^{2}+1) + \frac{20x * \cos(x)}{10x^{2}+1})$ .

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