Question

use your calculator to find the gradient of the graph of y = 2^{x} at x = 0. give your answer as a decimal, rounded to 3 significant figures.

Explanation:

Step1: Recall the derivative formula

The derivative of \(y = a^{x}\) is \(y^\prime=a^{x}\ln a\). For \(y = 2^{x}\), its derivative \(y^\prime=2^{x}\ln 2\).

Step2: Substitute \(x = 0\) into the derivative

When \(x = 0\), \(y^\prime|_{x = 0}=2^{0}\ln 2\). Since \(2^{0}=1\), then \(y^\prime|_{x = 0}=\ln 2\).

Step3: Calculate the value of \(\ln 2\) using a calculator

Using a calculator, \(\ln 2\approx0.693\) (rounded to 3 significant figures).

Answer:

\(0.693\)