Differentiating ex, In x and ax

Differentiating ex, In x and ax

Differentiating ex

  • The derivative of ex is ex. This applies to all bases of natural logarithms.

Differentiating In x

  • The derivative of In x is 1/x. This is important to remember as it is a specific rule and does not follow the general pattern of polynomial differentiation.

  • In differentiation, the In function refers to the natural logarithm. Natural logarithms have the base e.

Differentiating axe

  • If you are differentiating any exponent in the form axe, the rule to remember is to keep the function the same, but multiply by the natural log of a.

  • For example, if you are differentiating 2^x, the answer is 2^x * ln(2). The function remains the same, but is multiplied by the natural log of the base.

  • Remember that these rules apply when you are working with exponents that are not simply x, such as 3^x or 10^x.

Using Properties of Logarithms

  • Properties of logarithms can be used in differentiation to simplify the process.

  • For example, the property ln(a*b) = ln(a) + ln(b) can be used in differentiation to simplify a product into a sum. This means that d/dx[ln(x^2)] can be rewritten as d/dx[2*ln(x)] = d/dx[ln(x)] + d/dx[ln(x)].

  • These properties can often make differentiation easier by breaking down complex functions into simpler components.

Remember, it is paramount to practise these rules and concepts through examples and exercises in order to fully grasp the concept of differentiation and be proficient in its use.