Question

Find the derivative of y= 4tan5α

Answer 1

Given

$Y=4tan5\alpha............(1)$

We know that

$\frac{d}{da}(tanm\alpha)=sec^2(m\alpha).\frac{d}{d\alpha}(ma)\\=sec^2(m\alpha)=(mx_1)$

$\implies\frac{d}{d\alpha}(tanm\alpha)=msec^2(m\alpha)..................(2)$

Now differentiating (1) wrt $\alpha$ both sides we get

$\frac{dY}{d\alpha}=\frac{d}{d\alpha}(4tan5\alpha)=4.\frac{d}{d\alpha}(tan5\alpha)\\=4\times5sec^2(5\alpha): using(2)$

$\frac{dY}{d\alpha}=20sec^2(5\alpha)$