This web exercise gives practice applying differentiation rules to a variety of
differentiation problems.
All the calculus steps are shown in the solutions, but only minimal algebraic simplication is done.
If you want practice with the formulas only, then study
Differentiation Formula Practice.
There are two versions of this exercise—you may want to look at both to see which works best for you.
The other version has much stricter parenthesis usage.
to differentiate:  the basic rule  a more general rule  important things to remember 
power functions  the Simple Power Rule: $\displaystyle\frac{d}{dx} x^n = nx^{n1}$ 
the General Power Rule: $\displaystyle\frac{d}{dx} (f(x))^n = n\bigl(f(x)\bigr)^{n1}f'(x)$ 
power functions have a variable base, and a constant in the exponent; the Power Rules tell how to differentiate power functions 
exponential function, base $\text{e}$ 
$\displaystyle\frac{d}{dx} {\text{e}}^x = {\text{e}}^x$  $\displaystyle\frac{d}{dx} {\text{e}}^{f(x)} = {\text{e}}^{f(x)}\cdot f'(x)$  exponential functions have a constant base, and the variable in the exponent; for this special function, its derivative is itself—the $y$value of a point on the graph of $y = {\text{e}}^x$ is the same as the slope of the tangent line at the point 
exponential functions, base $a \gt 0$, $a\ne 1$ 
$\displaystyle\frac{d}{dx} a^x = a^x\ln a$  $\displaystyle\frac{d}{dx} a^{f(x)} = a^{f(x)}(\ln a)\cdot f'(x)$  exponential functions have a constant base, and the variable in the exponent; note that $\,\ln\text{e} = 1\,$, so the constant disappears when the base is the irrational number $\,\text{e}$ 
logarithmic function, base $\text{e}$ 
$\displaystyle\frac{d}{dx} \ln{x} = \frac{1}{x}$  $\displaystyle\frac{d}{dx} \ln f(x) = \frac{1}{f(x)}\cdot f'(x)$  the derivatives of logarithmic functions involve
the reciprocal of the input 
logarithmic functions, base $a \gt 0$, $a\ne 1$ 
$\displaystyle\frac{d}{dx} \log_a{x} = \frac{1}{x\ln a}$  $\displaystyle\frac{d}{dx} \log_a f(x) = \frac{1}{f(x)\ln a}\cdot f'(x)$  note that $\,\ln\text{e} = 1\,$, so the constant disappears when the base of the logarithm is the irrational number $\,\text{e}$ 
trigonometric functions, sine and cosine  $\displaystyle\frac{d}{dx} \sin x = \cos x$ $\displaystyle\frac{d}{dx} \cos x = \sin x$ 
$\displaystyle\frac{d}{dx} \sin f(x) = (\cos f(x))\cdot f'(x)$ $\displaystyle\frac{d}{dx} \cos f(x) = (\sin f(x))\cdot f'(x)$ 
sine and cosine are ‘cofunctions’: note the pattern: to find the derivative of a cofunction, replace each function in the derivative formula by its cofunction, and introduce a minus sign 
trigonometric functions, tangent and cotangent  $\displaystyle\frac{d}{dx} \tan x = \sec^2 x$ $\displaystyle\frac{d}{dx} \cot x = \csc^2 x$ 
$\displaystyle\frac{d}{dx} \tan f(x) = (\sec^2 f(x))\cdot f'(x)$ $\displaystyle\frac{d}{dx} \cot f(x) = (\csc^2 f(x))\cdot f'(x)$ 
tangent and cotangent are ‘cofunctions’: note the pattern: to find the derivative of a cofunction, replace each function in the derivative formula by its cofunction, and introduce a minus sign 
trigonometric functions, secant and cosecant  $\displaystyle\frac{d}{dx} \sec x = \sec x\tan x$ $\displaystyle\frac{d}{dx} \csc x = \csc x\cot x$ 
$\displaystyle\frac{d}{dx} \sec f(x) = (\sec f(x)\tan f(x))\cdot f'(x)$ $\displaystyle\frac{d}{dx} \csc f(x) = (\csc f(x)\cot f(x))\cdot f'(x)$ 
secant and cosecant are ‘cofunctions’: note the pattern: to find the derivative of a cofunction, replace each function in the derivative formula by its cofunction, and introduce a minus sign 
inverse trigonometric functions 
$\displaystyle\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1x^2}}$ $\displaystyle\frac{d}{dx} \arccos x = \frac{1}{\sqrt{1x^2}}$ $\displaystyle\frac{d}{dx} \arctan x = \frac{1}{1+x^2}$ 
$\displaystyle\frac{d}{dx} \arcsin f(x) = \frac{1}{\sqrt{1(f(x))^2}}\cdot f'(x)$ $\displaystyle\frac{d}{dx} \arccos f(x) = \frac{1}{\sqrt{1(f(x))^2}}\cdot f'(x)$ $\displaystyle\frac{d}{dx} \arctan f(x) = \frac{1}{1+(f(x))^2}\cdot f'(x)$ 
alternate notation: $\arcsin(x) = \sin^{1}(x)$ $\arccos(x) = \cos^{1}(x)$ $\arctan(x) = \tan^{1}(x)$ 
variable base to variable power 
Example (the ‘log trick’): Differentiate: $y = x^{2x}$


Because of the inverse relationship between $\ln x$ and ${\text{e}}^x\,$, we have: $${\text{e}}^{\ln x} = x\ \ \text{for all } x \gt 0$$ Also, a property of logarithms is: $$\ln x^y = y\ln x$$ 
In the exercises below, the calculus steps in finding the derivatives are shown.
However, only minimal simplification is done: