Master derivatives in CBSE Class 11 Applied Mathematics. Learn power rule, sum rule, and product rule with step-by-step solutions and practice problems.
The derivative of a function f(x) measures the instantaneous rate of change of the function with respect to x. It is defined as the limit of the difference quotient as the interval approaches zero. Geometrically, it represents the slope of the tangent line to the curve at any point x.
Step 1: Apply the definition f'(x) = lim(h→0) [f(x+h) − f(x)] / h.
Step 2: Since f(x) = 5, f(x+h) = 5.
Step 3: f'(x) = lim(h→0) [5 − 5] / h = lim(h→0) 0 / h = 0.
Answer: 0
For any real number n, the derivative of the power function xⁿ is given by multiplying the exponent by the base and subtracting one from the exponent. This is the most fundamental rule for differentiating polynomials. It applies to all integer and fractional powers of x.
Step 1: Identify n = 4.
Step 2: Apply the formula d/dx (xⁿ) = n·xⁿ⁻¹.
Step 3: f'(x) = 4·x⁴⁻¹ = 4x³.
Answer: 4x³
The derivative of a sum of two functions is the sum of their individual derivatives. Additionally, the derivative of a constant multiplied by a function is the constant times the derivative of the function. These rules allow us to differentiate complex polynomial expressions term by term.
Step 1: Differentiate each term separately.
Step 2: d/dx (3x²) = 3·(2x) = 6x.
Step 3: d/dx (2x) = 2·(1) = 2.
Answer: 6x + 2