Questions: Find the derivative of the following function. y=-5 x^2+7 cos x dy/dx=

Transcript text: Find the derivative of the following function. \[ \begin{array}{l} y=-5 x^{2}+7 \cos x \\ \frac{d y}{d x}=\square \end{array} \] $\square$

Solution

Solution Steps

To find the derivative of the given function, we need to apply the rules of differentiation. Specifically, we will use the power rule for the term involving $x^2$ and the derivative of the cosine function for the term involving $\cos x$.

Step 1: Define the Function

We start with the function given in the problem: \[ y = -5x^2 + 7\cos(x) \]

Step 2: Differentiate the Function

To find the derivative $\frac{dy}{dx}$, we apply the power rule and the derivative of the cosine function: \[ \frac{dy}{dx} = \frac{d}{dx}(-5x^2) + \frac{d}{dx}(7\cos(x)) \] Calculating each term, we have: \[ \frac{d}{dx}(-5x^2) = -10x \] \[ \frac{d}{dx}(7\cos(x)) = -7\sin(x) \] Thus, combining these results, we get: \[ \frac{dy}{dx} = -10x - 7\sin(x) \]