Question

differentiate.
f(x)=5x^9 - 3\cos(x)
f(x)=

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differentiate.
f(x)=4\sqrt{x}\sin(x)
f(x)=

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Explanation:

Step1: Differentiate term - by - term

We know that if \(y = ax^n\), then \(y^\prime=anx^{n - 1}\) and if \(y=\cos(x)\), then \(y^\prime=-\sin(x)\). For \(y = 5x^9-3\cos(x)\), we differentiate each part separately.

Step2: Differentiate \(5x^9\)

Using the power - rule \((x^n)^\prime=nx^{n - 1}\), for \(y = 5x^9\), we have \(y^\prime=5\times9x^{9 - 1}=45x^8\).

Step3: Differentiate \(-3\cos(x)\)

Since \((\cos(x))^\prime=-\sin(x)\), for \(y=-3\cos(x)\), we have \(y^\prime=-3\times(-\sin(x)) = 3\sin(x)\).

Step4: Combine the results

\(f^\prime(x)\) of \(f(x)=5x^9 - 3\cos(x)\) is \(f^\prime(x)=45x^8+3\sin(x)\).

for second function:

Step1: Use the product rule

The product rule states that if \(y = u(x)v(x)\), then \(y^\prime=u^\prime(x)v(x)+u(x)v^\prime(x)\). For \(f(x)=4\sqrt{x}\sin(x)\), let \(u(x)=4\sqrt{x}=4x^{\frac{1}{2}}\) and \(v(x)=\sin(x)\).

Step2: Differentiate \(u(x)\)

Using the power - rule \((x^n)^\prime=nx^{n - 1}\), for \(u(x)=4x^{\frac{1}{2}}\), we have \(u^\prime(x)=4\times\frac{1}{2}x^{\frac{1}{2}-1}=2x^{-\frac{1}{2}}=\frac{2}{\sqrt{x}}\).

Step3: Differentiate \(v(x)\)

Since \((\sin(x))^\prime=\cos(x)\), \(v^\prime(x)=\cos(x)\).

Step4: Apply the product rule

\(f^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)=\frac{2}{\sqrt{x}}\sin(x)+4\sqrt{x}\cos(x)\).

Answer:

\(45x^8 + 3\sin(x)\)