Questions: Find the derivative of (y=frac5x^3+7x^2x).

Find the derivative of (y=frac5x^3+7x^2x).
Transcript text: term denominators when possible. Find derivative 16. $y=\frac{5 x^{3}+7 x^{2}}{x}$
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Solution

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Solution Steps

To find the derivative of the given function, first simplify the expression by dividing each term in the numerator by the denominator. Then, apply the power rule of differentiation to each term.

Step 1: Simplify the Expression

The given function is \( y = \frac{5x^3 + 7x^2}{x} \). We can simplify this by dividing each term in the numerator by \( x \): \[ y = \frac{5x^3}{x} + \frac{7x^2}{x} = 5x^2 + 7x \]

Step 2: Differentiate the Simplified Function

To find the derivative of the simplified function \( y = 5x^2 + 7x \), we apply the power rule of differentiation. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

  • The derivative of \( 5x^2 \) is \( 10x \).
  • The derivative of \( 7x \) is \( 7 \).

Thus, the derivative of the function is: \[ \frac{dy}{dx} = 10x + 7 \]

Final Answer

\(\boxed{\frac{dy}{dx} = 10x + 7}\)

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