- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
Logarithmic Differentiation:
If function to the power function or function to the power variable or variable to the power function or variable to the power variable ,then apply logarithm.
The derivative of ex w.r.t. x = d/dx (ex) = ex
The derivative of log x w.r.t. x = d/dx (log x) = 1/x
Now this last limit
f'(x) = bxf'(0) = bx
So, in case of natural exponential functions, f(x) = ex
Note: In general exponential cases, for example, y = bx, where b is a real number. The derivative for this kind of function is
Question: Differentiate f(x) = 4ex – 5x
Answer:
The derivation of ex will remain ex, the derivative of 5x will become 5xln(5) as explained above.
Therefore, f'(x) = 4ex – 5xln(x)
Question : Find the value of F'(x) at x=0 when f(x) = 7x + 2ex
Answer:
Differentiating: f'(x) = 7xln(7) + 2ex
at x=0, f'(0) = 70ln(7) + 2e0
= ln(7) + 2
= 3.945
Question: d/dx(xx) = xx(1+ln x)
Question: Find the value of
if y = 2x{cos x}.
Solution: Given the function y = 2x{cos x}
Taking logarithm of both the sides, we get
log y = log(2x{cos x})
Now, differentiating both the sides w.r.t by using the chain rule we get,