Derivatives of logarithmic and exponential functions - Solved Examples

Example 1

`5ln(x) + 7e^x`

Solution:

`d/dx (5ln(x) + 7e^x)`

Take the derivative of the sum term by term,

`d/dx 5ln(x) + d/dx 7e^x`

Factor out the constants,

`5 d/dx ln(x) + 7 d/dx e^x`

Applying the formulas for logarithmic and exponential functions,

`5/x + 7e^x`

Example 2:

`x^2 / ln(x) + ln(x) / x^2`

Solution:

Take the derivative of the sum term by term,

`d/dx x^2/ln(x) + d/dx ln(x) / x^2`

Take the derivative of each quotient by the help of the quotient rule,

`(ln(x) d/dx x^2 - x^2 d/dx ln(x)) / ( ln(x) )^2 + (x^2 d/dx ln(x) - ln(x) d/dx x^2)/x^2`

Apply derivative rules (power rule, logarithmic/exponential functions derivatives)

`(ln(x) 2x - x^2 1/x) / ( ln(x) )^2 + (x^2 1/x - ln(x) 2x)/x^2`

Simplify the answer,

`(2xln(x) - x) / ( ln(x) )^2 - (2xln(x) - x)/x^2`

Example 3:

`log_5(x) + 2^x`

Solution:

Take the derivative of the sum term by term,

`d/dx log_5(x) + d/dx 2^x`

The formula for logarithmic differentiation for logarithms other than natural logarithms is `d/dx log_a(x) = 1/(x ln(a))`. In this question, the base of the logarithm, a is 5, and so its derivative is `1/(xln(5))`. The formula for exponential derivatives with base other than e is `d/dx a^x = a^x ln(a)`. In this question the base of the exponential function `2^x` is 2, and so its derivative is `2^x ln(2)`. Hence the derivative is,

`1/(xln(5)) + 2^xln(2)`

Example 4:

`e^(10x)`

Solution:

Since this exponential function has 10x as its exponent, and not just 'x', so the chain rule will be used to differentiate this function, as the differentiation rule for exponential functions applies only when the exponent is just 'x'.

By the chain rule, let u = 10x,

`d/dx e^(10x) = d/du e^u * d/dx u` ... where u is 10x,

Now since there is a single variable u in the exponential functoin, so applying the formula for differentiationg exponential functions,

`e^u d/dx u` ... where u is 10x,

Substituting back u = 10x,

`e^(10x) d/dx (10x)`

Derivative of 10x is 10,

`e^(10x) * 10 = 10e^(10x)`

Example 5:

`(10log_10(x))/(3log_3(x))`

Solution:

`d/dx (10log_10(x))/(3log_3(x))`

Factor out the constants from the derivative,

`10/3 d/dx (log_10(x))/(log_3(x))`

Applying the quotient rule,

`10/3 (log_3(x) d/dx log_10(x) - log_10(x) d/dx log_3(x))/((log_3(x))^2)`

Applying derivative formulas for logarithmic functions,

`10/3 ((log_3(x))/(xln(10)) - (log_10(x))/(xln(3)))/((log_3(x))^2)`