Calculating derivatives Find dy&gt;dx for the following functions. y=xsin⁡x

Step 1For a function f(x) the derivative is defined as f′(x)=limh→0f(x+h)−f(x)h. For some standard functions the derivative is known so if a function is in terms of standard functions then definition may not be used.The product rule of differentiation states that (f(x)g(x))′=f′(x)g(x)+f(x)g′(x). The derivative of xn,n≠0 is equal to nxn−1. The derivative of sin⁡x is cos⁡x.Step 2Given function is y=xsin⁡x. Use the product rule and other information from step 1 to find dydx.y=xsin⁡xdydx=ddx(xsin⁡x)=ddx(x)sin⁡x+xddx(sin⁡x)=(1)sin⁡xx+x(cos⁡x)=sin⁡x+xcos⁡xHence, derivative of given function is sin⁡x+xcos⁡x.

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Caelan

Answered question

2021-10-22

Calculating derivatives Find dy>dx for the following functions.

y = x \sin x

Answer & Explanation

Khribechy

Skilled2021-10-23Added 100 answers

Step 1
For a function f(x) the derivative is defined as $f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ . For some standard functions the derivative is known so if a function is in terms of standard functions then definition may not be used.
The product rule of differentiation states that $(f (x) g (x))^{'} = f^{'} (x) g (x) + f (x) g^{'} (x)$ . The derivative of $x^{n}, n \neq 0$ is equal to $n x^{n - 1}$ . The derivative of $\sin x i s \cos x$ .
Step 2
Given function is $y = x \sin x$ . Use the product rule and other information from step 1 to find $\frac{d y}{d x}$ .
$y = x \sin x$
$\frac{d y}{d x} = \frac{d}{d x} (x \sin x)$
$= \frac{d}{d x} (x) \sin x + x \frac{d}{d x} (\sin x)$
$= (1) \sin x x + x (\cos x)$
$= \sin x + x \cos x$
Hence, derivative of given function is $\sin x + x \cos x$ .

Jeffrey Jordon

Expert2022-08-15Added 2605 answers

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