Question

find the derivative of y=x4(5sinx-3cosx)​

Answer 1

Answer:

hope you can understand

Answer 2

As per tha data given in the above question .

We have to find the derivative of given question.

Now ,

By the differentiation we have to solve this above question.

Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables.

Solution:

$y = {x}^{4} (5 \: sin \: x - 3 \: cos \: x)$

Differentiate with respect to x ,

$\frac{dy}{dx} = \frac{d \: }{dx} ({x}^{4} (5 \: sin \: x - 3 \: cos \: x))$

By using product rule of derivative,

Let u = x⁴ and v= (5 sin x - 3 cos x)

$\frac{d}{dx}(uv) = \frac{d(u)}{dx} .v + u.\frac{d(v)}{dx} \: \: \: \: .....(1)$

Now put the values in (1) ,

$\frac{d}{dx}y = \frac{d( {x}^{4} )}{dx} . (5 \: sin \: x - 3 \: cos \: x) + \: {x}^{4} \times \frac{d(5 \: sin \: x - 3 \: cos \: x))}{dx}$

By using some formulaes,

$\frac{d}{dx} {x}^{n} = n \: {x}^{n - 1} \: \frac{d}{dx} x$

$\frac{d}{dx}sin \: x = cos \: x$]

$\frac{d}{dx}cos \: x = - sin \: x$

STEP-1

$\frac{d}{dx}y = ( 4{x}^{3} ). (5 \: sin \: x - 3 \: cos \: x) + \: {x}^{4} \times (5 \: cos \: x - 3 \: ( - sin \: x))$

STEP-2

$\frac{d}{dx}y = {x}^{3} (4. (5 \: sin \: x - 3 \: cos \: x) + \: {x}\times (5 \: cos \: x + 3 \: sin \: x)$

STEP-3

$\frac{d}{dx}y = {x}^{3} (20\: sin \: x - 12 \: cos \: x+ \: 5 x\: cos \: x + 3x \: sin \: x)$

Hence,

The derivate of y= x⁴(5 sinx -3 cos x) is

$\frac{d}{dx}y = {x}^{3} (20\: sin \: x - 12 \: cos \: x+ \: 5 x\: cos \: x + 3x \: sin \: x)$

Project code #SPJ2