Differentiation Free Online FlashCards

What is the limit definition of f'(x)?

The derivative is the limit f'(x)=lim_{h→0}[f(x+h)−f(x)]/h, when this limit exists as a finite number.

What is the relationship between differentiability and continuity?

If f is differentiable at a, then f is continuous at a. The converse is not always true; |x| is continuous but not differentiable at 0.

State the power rule.

For any real n, d/dx[x^n]=nx^(n−1), wherever the expression is defined.

How do you differentiate a product uv?

The product rule is d/dx[uv]=u'v+uv'.

State the quotient rule.

The quotient rule is d/dx[u/v]=(vu'−uv')/v², provided v(x)≠0.

What is the chain rule?

The chain rule is d/dx[f(g(x))]=f'(g(x))g'(x): differentiate the outer function, then multiply by the inner derivative.

What are the derivatives of sin x and cos x?

The derivative of sin x is cos x, while the derivative of cos x is −sin x; these formulas assume radians.

How do you differentiate e^{u(x)}?

For u=u(x), d/dx[e^u]=e^u u'. The exponential remains unchanged and is multiplied by the inner derivative.

What is the derivative of ln|u(x)|?

For u(x)≠0, d/dx[ln|u(x)|]=u'(x)/u(x). For x>0, this gives d/dx[ln x]=1/x.

What is the derivative of log_a x?

For x>0, d/dx[log_a x]=1/(x ln a), where a>0 and a≠1.

What is the derivative of x^x?

For x>0, the derivative of x^x is x^x(ln x+1), found by logarithmic differentiation.

State the derivative formula for an inverse function.

If f is one-to-one and differentiable, then (f⁻¹)'(x)=1/f'(f⁻¹(x)), provided the denominator is nonzero.