Question

differeniate the following w.r.t. x by using these formulae

Answer 1

Here is your answer...

Answer 2

To Find:

$\frac{d}{dx} [ sin {}^{m} x.cos {}^{n} x]$

Solution:

On applying the Product rule:

$\frac{d}{dx}[f {} {1} (x).f {2} (x)] \\ = f1(x). \frac{d}{dx} [f2(x)] + f2(x). \frac{d}{dx} [f1(x)]$

Thus,

$\frac{d}{dx} [ \sin ^{m} (x) . \cos ^{n} (x) ]$

is given by:

$= \sin {}^{m} (x) .[- sin(x).n.cos {}^{n - 1} (x)] + cos(x).[m.sin {}^{m - 1} (x).cos(x)]$

$= m. \cos {}^{n + 1} (x) . \sin {}^{m - 1} (x) - n. \sin {}^{m + 1} (x) . \cos {}^{n - 1} (x)$

Therefore,

The derivative of [sin^mx.cosⁿx] is given by:

$m. \cos {}^{n + 1} (x) . \sin {}^{m - 1} (x) - n. \sin {}^{m + 1} (x) . \cos {}^{n - 1} (x)$