1) $cos(10x-5)$
outer: $cos()$ and its derivative: $-sin()$
plugging in the argument $\mathbf{-sin(10x-5)}$
inner: $10x-5 $, and its derivative: 10
final derivative:
$(cos(10x-5))'$=$ -sin(10x-5)\cdot 10 $
2) $sin(15x)$
outer: $sin()$ and its derivative: $cos()$
plugging in the argument into derivative
$\mathbf{cos(15x)}$
inner: $15x $, and its derivative: 15
final derivative:
$(sin(15x))'$=$ cos(15x)\cdot 15 $
3) $sin(x-9)$
outer: $sin()$ and its derivative: $cos()$
plugging in the argument into derivative
$\mathbf{cos(x-9)}$
inner: $x-9 $, and its derivative: 1
final derivative:
$(sin(x-9))'$=$ cos(x-9)\cdot 1$
4) $(x^3-2x)^2$
outer: $()^2$ and its derivative: $2()^1$
plugging in the argument into derivative
$\mathbf{2(x^3-2x)}$
inner: $x^3-2x $, and its derivative: $3x^2-2$
final derivative:
$((x^3-2x)^2)'$=$ 2(x^3-2x)\cdot (3x^2-2)$
5) $\sqrt{-x-3}$
outer: $\sqrt{()}$
inner: $-x-3 $
6) $\frac{5}{(3x)^3}$
we can write:
$\frac{5}{(3x)^3}=5\cdot (3x)^{-3}$
outer: $5\cdot ()^{-3}$
inner: $3x $
7) $\frac{2}{\sqrt{5x}}$
we can write:
$\frac{2}{\sqrt{5x}}=2\cdot (5x)^{-\frac{1}{2}}$
outer: $2\cdot ()^{-\frac{1}{2}}$
inner: $5x $
8) $e^{x^4+2}$
outer: $e^{()}$
inner: $x^4+2$
9) $2ln(2x-10)$
outer: $2ln()$
inner: $2x-10$
10) $e^{-x}$
outer: $e^{()}$
inner: $-x$
11) $2e^{3x^5-x+1}$
outer: $2e^{()}$ and its derivative: $2e^{()}$
plugging in the argument into derivative
$\mathbf{2e^{(3x^5-x+1)}}$
inner: $3x^5-x+1 $, and its derivative: $15x^4-1$
final derivative:
$(2e^{3x^5-x+1})'$=$ 2e^{(3x^5-x+1)}\cdot (15x^4-1)$
12) $10e^{x^5-2x^4}$
outer: $10e^{()}$ and its derivative: $10e^{()}$
plugging in the argument into derivative
$\mathbf{10e^{x^5-2x^4}}$
inner: $x^5-2x^4 $, and its derivative: $5x^4-8x^3$
final derivative:
$(10e^{x^5-2x^4})'$=$ 10e^{x^5-2x^4}\cdot (5x^4-8x^3)$
13) $\frac{11}{ln(x)}$
we can write:
$\frac{11}{ln(x)}=11\cdot (ln(x))^{-1}$
outer: $11()^{-1}$ and its derivative: $-11()^{-2}$
plugging in the argument into derivative
$\mathbf{-11(ln(x))^{-2}}$
inner: $ln(x)$, and its derivative: $\frac{1}{x}$
final derivative:
$\Big(\frac{11}{ln(x)}\Big)'$=$ -11(ln(x))^{-2}\cdot \frac{1}{x}$
14) $sin(cos(x))$
outer: $sin()$ and its derivative: $cos()$
plugging in the argument into derivative
$\mathbf{cos(cos(x))}$
inner: $cos(x)$, and its derivative: $-sin(x)$
final derivative:
$(sin(cos(x)))'$=$ cos(cos(x))\cdot (-sin(x))$
15) $ln(sin(x))$
outer: $ln()$ and its derivative: $\frac{1}{()}$
plugging in the argument into derivative
$\mathbf{\frac{1}{(sin(x))}}$
inner: $sin(x)$, and its derivative: $cos(x)$
final derivative:
$(sin(cos(x)))'$=$ \frac{1}{(sin(x))}\cdot cos(x)$
16) $e^{cos(ln(x))}$
first inner: $ln(x)$ and its derivative: $\frac{1}{x}$
you may say that first inner function is "x", but its derivative is one,
so it doesn't make difference as we would multiply by 1. So, we skip it.
second inner: $cos()$, and its derivative: $-sin(ln(x))$
outer: $e^{()}$ with derivative $e^{cos(ln(x))}$
final derivative:
$(e^{cos(ln(x))})'=-sin(ln(x))\frac{e^{cos(ln(x))}}{x}$
17) $cos(sin(15x))$
first inner: $15x$ and its derivative 15
second inner: $sin()$ and its derivative: $cos(15x)$
outer: $cos()$ with derivative $-sin(sin(15x))$
final derivative:
$(cos(sin(15x)))'=-15cos(15x)sin(sin(15x))$
18) $sin(sin^2(x+2))$
first inner: $sin(x+2)$ and its derivative $cos(x+2)$
second inner: $()^2$ and its derivative: $2sin(x+2)$
outer: $sin()$ with derivative $cos(sin^2(x+2))$
final derivative:
$(sin(sin^2(x+2)))'=\\ =cos(x+2)2sin(x+2)cos(sin^2(x+2))$
19) $tan(cos^2(x))$
first inner: $cos(x)$ and its derivative $-sin(x)$
second inner: $()^2$ and its derivative: $2cos(x)$
outer: $tan()$ with derivative $sec^2(cos^2(x))$
final derivative:
$(tan(cos^2(x)))'=\\ =-2sin(x)cos(x)sec^2(cos^2(x))$
20) $log_8 e^{2x}$
first inner: $2x$ and its derivative 2
second inner: $e^{()}$ and its derivative: $e^{2x}$
outer: $log_8 ()$ with derivative $\frac{1}{e^{2x}ln(8)}$
final derivative:
$(log_8 e^{2x})'=\\ =\frac{2e^{2x}}{e^{2x}ln(8)}=\frac{2}{ln(8)}$
21) $log_5 ^2 (sin(x))$
first inner: $sin(x)$ and its derivative $cos(x)$
second inner: $log_5 ()$ and its derivative: $\frac{1}{sin(x)ln(5)}$
outer: $()^2$ with derivative $2log_5 (sin(x))$
final derivative:
$(log_5 ^2 (sin(x)))'=\\ =\frac{2cos(x)log_5 (sin(x))}{sin(x)ln(5)}$