Rules On Differentiation
Product Rule:
$$\frac{d}{dx} \left( uv \right) = v \frac{du}{dx} + u \frac{dv}{dx}$$
,where u, v are functions of x
Quotient Rule:
$$\frac{d}{dx} \left(\frac{u}{v} \right) = \frac{v\frac{du}{dx} – u \frac{dv}{dx}}{v^{2}}$$
Chain Rule:
$$\frac{d}{dx} \left[ f\left(u \right) \right] = \frac{d}{du} \left[ f \left( u \right) \right] \times \frac{du}{dx}$$
Integration By Parts:
$$\int \! u \, \text{d}v = uv-\int \! v \, \text{d}u$$
Basic Properties of Derivatives
$$\frac{d}{dx}(cf(x)) = cf'(x)$$
$$\frac{d}{dx} (x^{n}) = nx^{n-1}$$
$$(fg)’ = f’g + fg’$$
$$\frac{d}{dx} (f(g(x))) = f'(g(x))g'(x)$$
$$\frac{d}{dx} \left( ln \, g(x) \right) = \frac{g'(x)}{g(x)}$$
$$(f(x) \pm g(x))’ = f'(x) \pm g'(x)$$
$$\frac{d}{dx}(c) = 0$$
$$\left( \frac{f}{g} \right)’ = \frac{f’g \, – fg’}{g^{2}}$$
$$\frac{d}{dx} \left( e^{g(x)} \right) = g'(x) e^{g(x)}$$
Note: c is any constant, n is any number
Standard Derivatives:
Polynomials
$$\frac{d}{dx} (c) = 0$$
$$\frac{d}{dx}(x) = 1$$
$$\frac{d}{dx} (cx) = c$$
$$\frac{d}{dx} \left( x^{n} \right) = nx^{n-1}$$
$$\frac{d}{dx} \left( cx^{n} \right) = ncx^{n-1} $$
Trig. Functions
$$\frac{d}{dx} (\sin \, x) = \cos \, x$$
$$\frac{d}{dx} (\cos \, x) = \, – \sin \, x$$
$$\frac{d}{dx} (\tan \, x) = \sec^{2} \, x$$
$$\frac{d}{dx} (\sec \, x) = \sec \, x \, \tan \, x$$
$$\frac{d}{dx} (\csc \, x) = \, – \csc \, x \, \cot \, x$$
$$\frac{d}{dx} (\cot \, x) = \, – \csc^{2} \, x$$
Inverse Trig. Functions
$$\frac{d}{dx} (\sin^{-1} \, x) = \frac{1}{\sqrt{1 \, – x^{2}}}$$
$$\frac{d}{dx} (\cos^{-1} \, x) = \, – \frac{1}{\sqrt{1 \, – x^{2}}}$$
$$\frac{d}{dx} (\tan^{-1} \, x) = \frac{1}{1 + x^{2}}$$
$$\frac{d}{dx} (\sec^{-1} \, x) = \frac{1}{ |x| \sqrt{x^{2}-1}}$$
$$\frac{d}{dx} (\csc^{-1} \, x) = \, – \frac{1}{ |x| \sqrt{x^{2}-1}}$$
$$\frac{d}{dx} (\cot^{-1} \, x) = \, – \frac{1}{1+ x^{2}}$$
Exponential/Logarithm Functions
$$\frac{d}{dx} \left( a^{x} \right) = a^{x} \, ln \,(a)$$
$$\frac{d}{dx} \left( e^{x} \right) = e^{x}$$
$$\frac{d}{dx} (ln (x)) = \frac{1}{x}, \, \, x \gt 0$$
$$\frac{d}{dx} ( ln |x|) = \frac{1}{x}, \, \, x \ne 0$$
$$\frac{d}{dx} (log_{a} \, (x)) = \frac{1}{x \, ln \, a}, \, \, x \gt 0$$
Hyperbolic Trig. Functions
$$\frac{d}{dx} (\sinh \, x) = \cosh \, x$$
$$\frac{d}{dx} (\cosh \, x) = \sinh \, x$$
$$\frac{d}{dx} (\tanh \, x) = \text{sech}^{2} \, x$$
$$\frac{d}{dx} (\text{sech} \, x) = \, – \text{sech} \, x \, \tanh \, x$$
$$\frac{d}{dx} (\text{csch} \, x) = \, – \text{csch} \, x \, \coth \, x$$
$$\frac{d}{dx} (\coth \, x) = \, – \text{csch}^{2} \, x$$
