Derivative of Sec Square x

The derivative of sec square x is equal to twice the product of sec square x and tanx. Mathematically, we can write the derivative of sec^2x as d(sec^2x)/dx = 2 sec²x tanx. We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits. Differentiation of a function gives the slope function of the curve of the function which can give the slope of the function at a particular point.

Let us learn to evaluate the derivative of sec square x using different methods of derivatives and its formula. We will also solve different examples related to the concept for a better understanding of the concept of derivatives.

The derivative of sec^2x is equal to 2 sec²x tanx. It is mathematically written as d(sec^2x)/dx = 2 sec²x tanx. Sec x is one of the important and main trigonometric functions in trigonometry. We know that the derivative of secx is equal to secx tanx. We can use the power rule of differentiation and the derivative of sec x to find the derivative of sec square x. In the next section, let us go through the formula for the differentiation of sec^2x.

The formula for the derivative of sec^2x is given by d(sec^2x)/dx = 2 sec^2x tanx. We know that the power rule of differentiation is d(x^n)/dx = n x^(n-1), that is, d(xⁿ)/dx = nx^n-1. Using this formula, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec²x tanx. The image below shows the formula for the derivative of sec square x:

The chain rule of derivatives is used to find the derivatives of composite functions. Sec^2x is the composite function of two functions f(x) = x² and g(x) = secx given by, h(x) = f o g (x) = sec^2x. Now, to find the derivative of sec^2x, we use the formula of chain rule given by, h'(x) = [f(g(x))]' = f'(g(x)) × g'(x). Hence, using this formula, the power rule of derivatives and the derivative of secx, we have

d(sec^2x)/dx = 2 sec^2-1x × d(secx)/dx

= 2 secx × secx tanx

= 2 sec²x tanx

Hence, the derivative of sec square x is equal to 2 sec²x tanx.

Now that we know that the derivative of sec^2x is equal to 2 sec²x tanx, we will prove this using the quotient rule of derivatives. We know that sec x is the reciprocal identity of cos x, that is, secx = 1/cosx. Also, the formula for the quotient rule of differentiation is (f/g)' = (f'g - fg')/g². Using these formulas, we have

d(sec²x)/dx = d(1/cos²x)/dx

= (1/cos²x)'

= [(1)' cos²x - 1 (cos²x)']/cos⁴x

= (0 × cos²x + 2 cosx sinx)/cos⁴x --- [Because the derivative of cosx is equal to -sinx]

= (2 cosx sinx)/cos⁴x

= 2 sinx/cos³x

= 2 (1/cos²x) (sinx/cosx)

= 2 sec²x tanx

Hence, we have proved that the derivative of sec square is equal to 2 sec²x tanx.

The product rule of differentiation is used to find the derivative of the product of two functions. For a function h(x) = f(x) g(x), the formula for its derivative is given by h'(x) = f'(x) g(x) + f(x) g'(x). For the function sec square x, we can writ it as h(x) = sec²x = secx × secx. Therefore, to find the derivative of sec^2x, we have

d(sec²x)/dx = (secx × secx)'

= (secx)' secx + secx (secx)'

= secx tanx secx + secx secx tanx --- [Because the derivative of secx is equal to secx tanx]

= 2 sec²x tanx

Hence, the derivative of sec^2x is equal to 2 sec²x tanx.

Important Notes on Derivative of Sec Square x

• The derivative of sec^2x is equal to 2 sec²x tanx.
• The formula for the derivative of sec square x is d(sec²x)/dx = 2 sec²x tanx
• We can find the differentiation of sec square x using different methods of differentiation such as product rule, quotient rule, and chain rule.

☛ Related Topics:

• Integration of 2x
• Derivative of 2x
• Integral of 2 sinx

What is Derivative of Sec Square x?

The derivative of sec square x is equal to twice the product of sec square x and tanx. Mathematically, we can write the derivative of sec^2x as d(sec^2x)/dx = 2 sec²x tanx.

What is the Formula for the Derivative of Sec^2x?

The formula for the derivative of sec^2x is given by d(sec²x)/dx = 2 sec²x tanx. Using the formula for the power rule of derivatives, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec²x tanx.

How Do You Prove the Derivative of Sec Square x?

We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits.

What is the Derivative of Sec Square x Square w.r.t. x Square?

The derivative of sec square x square with respect to x square is equal to 2 sec²(x²) tan(x²). We can find this using the chain rule method of differentiation.

How to Find the Second Derivative of Sec^2x?

To find the second derivative of sec^2x, we will differentiate its first derivative. We know that the derivative of sec^2x is equal to 2 sec²x tanx. Differentiating this gives us the second derivative of sec square x. The second derivative of sec square x is 2 sec²x [2 tan²x + sec²x].

What is the Antiderivative of Sec^2x?

The antiderivative of sec^2x is equal to tanx + C, where C is the integration constant. The antiderivative of a function is nothing but its integral and we know that the derivative of tanx is equal to sec²x. Therefore, the antiderivative of sec square x is tanx + C.