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:

$\frac{d}{dx}f(g(x))=f^{\prime}(g(x))g^{\prime}(x)$

Then,

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

$\frac{d}{dx}[e^{ax\^6}6}]=e^{ax\^6}6}\cdot\frac{d}{dx}[ax^6]=e^{ax\^6}6}\cdot6ax^5$

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

$\frac{d}{dx}\sin(ax^8)=\cos(ax^8)\cdot\frac{d}{dx}(ax^8)=\cos(ax^8)\cdot8ax^7$

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

$\frac{d}{dx}\cos(ax^8)=-\sin(ax^8)\frac{d}{dx}(ax^8)=-\sin(ax^8)8ax^7$

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

$\frac{d}{dx}\tan(ax^7)=\sec^2(ax^7)\cdot\frac{d}{dx}(ax^7)=\sec^2(ax^7)\cdot7ax^6$