Question

3. (5x)3 cos 2x Diffrentiate please​

Answer 1

Correct Question :

$\tt Differentiate \: {(5x) }^{3 \cos 2x} w.r.t.x.$

Solution :

Let try to differentiate w.r.t.x

We have,

$\tt\dashrightarrow y = {(5x) }^{3 \cos 2x}$

Taking both sides Log,

$\tt\dashrightarrow log \: y = log \: {(5x) }^{3 \cos 2x}$

$\tt\dashrightarrow log \: y = 3 \cos 2x. log \: 5x.$

Then,

Let Apply UV method or, ( Product method )

When,

• y = UV.
• dy/dx = U dv/dx + V du/dx.

$\tt\dashrightarrow \frac{1}{y} \frac{dy}{dx} = 3[ \cos \: 2x. \frac{1}{5x} \times 5 + log5x \times ( - \sin 2x) \times 2 ]$

$\tt\dashrightarrow \frac{dy}{dx} = 3y[ \frac{ \cos 2x }{x} - 2 \sin 2x. log 5x.]$

Putting the value of y in above equation, We get.

$\tt\dashrightarrow \frac{dy}{dx} = 3{(5x) }^{3 \cos 2x } [\frac{ \cos 2x }{x} - 2 \sin 2x. log 5x.]$

Therefore, the value of differentiation be 3(5)^3 cos 2x [cos 2x/ x -2 sin 2x log 5x.]

$\rule{200}1$

Some information :

$\tt\bullet\: {log \:m^{n}} = {n \: log \:m}$

When, y = x².

• dy/dx = nx^n-1.

When, y = 5x³.

• dy/dx = 15x².

When, y = 2x⁴.

• dy/dx = 8x³.