Maths WorkshopMaths problems with full solutions

Differentiable functions

20 problems · Differentiable and continuously differentiable functions.

Ex.8Polynomial bounds for sine and cosine
#inequalities #monotonicity
Differentiable functions
L1
L1
Ex.203Derivative of a quotient
#derivative #quotient rule
Differentiable functions
High school
High school
Ex.213Derivative and tangent line of a cubic
#derivative #tangent line #polynomials
Differentiable functions
High school
High school
Ex.214The derivative from the definition
#definition of the derivative #difference quotient #square root
Differentiable functions
High school
High school
Ex.362Derivative of an inverse function
#inverse function #implicit differentiation #arcsine
Differentiable functions
L1
L1
Ex.363A surprisingly simple derivative
#chain rule #composition #square roots
Differentiable functions
L1
L1
Ex.364Derivative with an absolute value
#differentiability #absolute value #one-sided derivatives
Differentiable functions
L1
L1
Ex.365Differentiability when gluing two pieces
#differentiability #piecewise functions #one-sided derivatives
Differentiable functions
L1
L1
Ex.376Logarithmic differentiation
#derivative #exponential form #powers with variable exponent
Differentiable functions
L1
L1
Ex.387Tangent lines through an external point
#tangent line #parabola #geometry
Differentiable functions
High school
High school
Ex.398Three ways to fail to be differentiable
#differentiability #corner #vertical tangent #oscillation
Differentiable functions
L1
L1
Ex.760Functions whose derivative is proportional to themselves
#exponential function #differential equation #functional equations
Differentiable functions
L1
L1
Ex.764L'Hôpital's rule, twice
#L'Hôpital's rule #indeterminate forms
Differentiable functions
L1
L1
Ex.789The n-th derivative of 1/(x² − 1)
#higher derivatives #partial fractions
Differentiable functions
L1
L1
Ex.790The n-th derivative of x² e^x
#higher derivatives #Leibniz formula #induction
Differentiable functions
L1
L1
Ex.795When L'Hôpital's rule gives no answer
#L'Hôpital's rule #counterexamples #squeeze theorem
Differentiable functions
L1
L1
Ex.1469A functional equation solved with derivatives
#functional equations #differentiation #polynomials
Differentiable functions
Olympiad
Olympiad
Ex.1480A convex function and its asymptote
#convexity #slopes #asymptotes
Differentiable functions
L1
L1
Ex.1481Convexity forces the derivative to be continuous
#convexity #Darboux theorem #monotone functions #continuity
Differentiable functions
L1
L1
Ex.1522Differentiable but not continuously differentiable
#differentiability #continuity of the derivative #counterexamples
Differentiable functions
L1
L1