Find first desivatine 3ex-1/5x+2x-sec x

Question

Find first desivatine
3e^x−1/5x+2^x−sec x

Answer 1

3e^x - 1/5 x + 2^x - sec x

 

This problem requires you to recall several derivative rules.

 

1) d/dx (3e^x) = 3e^x

2) d/dx (1/5 x) = 1/5

 

3) d/dx (2^x)

This one's a bit more complex

Let's rewrite 2x as (e^ln2)^x

This works because e^ln2 = 2

 

Working with exponents we know that (a^b)^x = a^bx, so (e^ln2)^x = e^(ln2)x

We have this in the form of e^x and we can use the chain rule, recognizing that ln2 is the coefficient of x

d/dx e^(ln2)x = e^(ln2)x ln2

= (e^ln2)^x ln2 recall that e^ln2 = 2 so

=2^x ln2

 

4) d/dx sec x

=(sec x)(tan x)

You can do this by memorizing the derivatives of trig functions or by realizing that sec x = 1/cos x and using the quotient rule.

 

We've not differentiated each piece

 

d/dx (3e^x-1/5x+2^x-sec x) = 3e^x - 1/5 + 2^x ln2 - sec(x) tan(x)

 

Hope this helps.