derivative of f(x)=(7x^2+7x+7)/(sqrt(x))
We save the definition:
1) Let's perform the differentiation:
For = and = , use the differentiation rule:
2) Let's differentiate :
ⓘ The derivative of a sum is the sum of the derivatives: =
ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u': = , = ■The derivative of x^n is n*x^(n-1): = ■Adding 0 changes nothing: =
ⓘ Multiplications are done before additions and subtractions: =
3) Let's differentiate :
ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)): =
4) We evaluate the resulting derivative ()':
ⓘ Multiplying a fraction by a term multiplies its numerator: = = ■To add fractions, put them over the same denominator, then add the numerators: = ■A power of a power multiplies the exponents, (a^n)^m = a^(n*m): =
ⓘ A minus sign before a parenthesis changes the sign of each of its terms: = ■Multiplying powers with the same base adds the exponents: = ■Multiplying a fraction by a number multiplies its numerator: = = ■We simplify the fraction by dividing the numerator and the denominator by , their greatest common divisor: =
ⓘ Dividing a fraction by is multiplying its denominator by : =
ⓘ Multiplying powers with the same base adds the exponents: =
ⓘ To add fractions, put them over the same denominator, then add the numerators: =
ⓘ The numerator and the denominator are computed before the division: =
ⓘ Expand by multiplying each term inside the parentheses: =
ⓘ Expand by multiplying each term inside the parentheses: =
ⓘ Like terms are grouped by adding their coefficients: = , =
5) Then you can resume the differentiation:
ⅰ =
▶Derivative:
