differentiate.\nf(x)=5x^9 - 3\\cos(x)\nf(x)= \n\nresources\nread it watch it\n\nsubmit answer\n\n15. - / 1…

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)).

Answer:

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

Explanation 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:

(\frac{2\sin(x)}{\sqrt{x}}+4\sqrt{x}\cos(x))