Question

Please help me work through these or if thats too much, "(b)" thank you!

Answer 1

In all these derivatives, we need to use the chain rule of derivatives. The rule tell us, that for two function f and g, the derivative of their composition is:

`(d)/(dx)f(g(x))=f^(\prime)(g(x))g^(\prime)(x)`

Then,

a) Here we know that the derivative of the exponential function, is the exponential function. By the chain rule:

`(d)/(dx)[e^(ax\^6)6}]=e^(ax\^6)6}\cdot(d)/(dx)[ax^6]=e^(ax\^6)6}\cdot6ax^5`

b) The derivative of the sin function is the cosine function. Again, by the chain rule:

`(d)/(dx)\sin(ax^8)=\cos(ax^8)\cdot(d)/(dx)(ax^8)=\cos(ax^8)\cdot8ax^7`

c) The derivative of the cosine function is minus the sine function:

`(d)/(dx)\cos(ax^8)=-\sin(ax^8)(d)/(dx)(ax^8)=-\sin(ax^8)8ax^7`

d) The derivative of the tangent is the secant squared.

`(d)/(dx)\tan(ax^7)=\sec^2(ax^7)\cdot(d)/(dx)(ax^7)=\sec^2(ax^7)\cdot7ax^6`